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So long as you're given two points, you can write a liner equation in just a few quick moments. Write linear equations given two points with help from an experienced math teacher in this free video clip.
Adding linear equations requires you to keep a few very important things in mind. Learn how to add linear equations with help from an experienced math teacher in this free video clip.
Linear equations need to be written in a very specific way for maximum accuracy. Write linear equations in algebra with help from an experienced math teacher in this free video clip.
Forming linear equations from a table of values is something that you have to do in a very specific way. Find out how we form linear equations from a table of values with help from a high school math tutor in this free video clip.
Determining if each set of points graphs a linear equation is something that you can do by breaking the process down into a series of basic, manageable steps. Determine if each set of points graphs a linear equation with help from an experienced mathematics professional in this free video clip.
Writing an equation requires you to keep a few key things in mind at all times. Learn about how to determine if a relationship in a table is linear and how to write the associated equation is with help from an experienced mathematics professional in this free video clip.
A linear equation is normally expressed via an equation. Find out about linear relationships between variables with help from an experienced mathematics professional in this free video clip.
Linear equations are a necessary part of mathematics for a variety of different reasons. Find out why a linear equation is needed with help from a mathematics tutor and educator in this free video clip.
Certain algorithms are great for expressing linear equations. Learn about algorithms for linear equations with help from a mathematics tutor and educator in this free video clip.
When graphing a liner equation you will also need to find the points, called intercepts, where the line intersects with the axes. Graph a linear equation by finding the intercepts with help from an experienced math teacher in this free video clip.
While graphing you may notice that a linear equation never seems to curve. Learn why a linear equation never curves when graphing with help from an experienced math teacher in this free video clip.
How you will rewrite a liner equation like Ax + B depends on the specifics of the equation in front of you. Rewrite the linear equation Ax + B with help from an experienced math teacher in this free video clip.
The slope intercept form of an equation is expressed as Y = MX + B. Learn the explanation of a linear equation and the slope effect with help from an experienced math teacher in this free video clip.
A non-linear equation typically has a few common characteristics. Learn about non-linear equations with help from an experienced math tutor in this free video clip.
Simple linear equations are defined in a very specific way. Learn about the definition of simple linear equations with help from an experienced math tutor in this free video clip.
Working with linear equations requires you to keep a few rules in mind. Learn about the rules for linear equations with help from an experienced math tutor in this free video clip.
Expressing a linear equation using your keyboard is the first step in learning to express yourself algebraically on the Internet. For starters, linear equations consist of coefficients/constants, variables, an equal sign and arithmetic operations like addition, subtraction, multiplication and division. Not included in linear equations are square roots and exponents. Furthermore, variables can only be divided by constants.
Students typically learn to solve systems of linear difference equations during the latter part of a high school algebra course. Also sometimes referred to as "simultaneous equations," systems of equations can pose a challenge for many students. The topic, however, offers students an opportunity for flexibility and creativity rarely seen in basic algebra because they're allowed to choose their own method of solving the problem.
To transform a table of data into a linear equation, you need to understand the dynamics of a linear equation, which is defined mathematically as y = mx + b. X is a variable that represents one column of data, y is a variable that represents another column of data, m is the slope (rise over run) and b is the y-intercept (where x equals 0). Realize also that one row of data (x | y) represents a point in the form (x, y). Knowing these basic properties of a linear equation, you can quickly transform any table or tables…
Graphing proportionate relationships between two quantities is no different than graphing a linear equation. For reference, a linear equation is a relationship with two variables or quantities that vary depending on each other's value. A typical linear equation is in the form y = mx + b, but a proportionate linear equation is in the form y = mx. As the value of x changes, so does the value of y. To graph such a relationship, you need only find the intercepts and then use the slope to plot additional points.
Before you can move onto the world of parabolas, hyperbolas and ellipses, you must first understand and know how to graph a linear equation. For clarification, such an equation features two variables, both of which are raised to the first power; y = 2x + 3 is a linear equation, and so is 43 = 2x + 2y. However, y = x^2 is not a linear equation. One way to graph a linear equation is by using what's known as a table of values.
Learning about linear equations doesn't have to be a tedious process. When you incorporate things that eighth-graders find entertaining, such as cartoons and games, learning about linear equations can become easier and much more pleasant. Many kids find it easier to understand abstract ideas and concepts when the learning is hands on, or in real life situations, and these projects allow them to do so.
Scientist and engineers use matrices in all forms of work. The technique is key in extending numerical analysis to vast data arrays--calculation of which would be impossible with conventional linear number sets. Matrices reduce the complexity of the high-level math involved, allowing mathematical operations to be applied to the matrix as a whole rather than tediously working through element by element. Matrices are hindered by their simplicity however. They are not suitable as a form of presentable information to another person, and require tools such as MATLAB to transform the raw data into an understandable context.
Boolean algebra is the logical and mathematical formula of cause and effect. Created by George Boole in the mid 1800s, the concept is the basis of the binary computer language but people can use this for everyday problem solving. In fact, many people use it everyday without ever even knowing it.
A Boolean expression is an algebraic expression that results in one of two possible values, 1 ("true") or 0 ("false"), known as Boolean values. Boolean logic forms the basis of calculations in modern binary, or base two, computer systems. You can use a system of Boolean expressions to represent any electronic computer circuit.
A simple linear equation contains only one variable and numbers. The purpose of the equation is to use algebra to move numbers away from the variable until it is isolated on one side of the equals sign, thus providing a value for the unknown. A one-step linear equation only requires one algebraic action for the solution. For example, 2x = 10 is a one-step equation because dividing both sides by 2 results in x = 5. Equations with multiple steps are easy to solve as long as the proper steps are followed.
Utilizing the Six Sigma method of calculating bias in clinical chemistry provides highly accurate reference data to determine the degree of deviance in a method from its goal and allows researchers to decide if test outcomes yield reliable results based on whether its total error falls within the tolerable limits. This process should be applied to data yielded from a variety of clinical chemistry tests, including creatinine clearance, plasma osmolality and urea reduction ratio.
Used to simplify the statements made in Boolean expressions, Boolean theorems use alphabetic letters like x, y and z to represent statements of truth or falsehoods. These theorems are then used to govern the possible relationships that these values can have, assuming each one only represents the value of 1 (true) or 0 (false).
Motion is a phenomenon that is best explained by a variety of physics-related theories. Linear motion, as its name suggests, is the motion of an object in a single direction. An object in linear motion will remain that way unless a certain type of force acts on it.
Linear equations contain variables, or letter representations for unknown quantities, and constants (numbers). The graph of a linear equation forms a straight line defined by its slope. The slope is the distance between a point, (x1, y1), and the point next to it, (x2, y2). The slope is defined as (y2 - y1) / (x2 - x1). The slope and the y-intercept, or point at which the line crosses the y-axis, are integral parts of the slope intercept form. Slope intercept form states y = mx + b, where "m" is the slope and "b" is the y-intercept.
A paraboloid is the surface of revolution of a parabola; that is, it is the shape formed when a parabola is revolved around a third axis. The surface of the reflectors in the headlights of cars are paraboloids. Paraboloids can be expressed by the formula z = b(x^2 + y^2) where b is a constant and x and y are variables. You can use Excel to make a graph of a paraboloid.
Solving mathematical equations often involves applying the opposite of a number or operation to cancel it out in order to get the variable isolated on one side of the equation. The opposite operation of addition is subtraction and the opposite of multiplication is division. But the identity properties and inverse properties present additional opposites that present not a method of elimination, but a method of solving for a desired answer.
Linear equations have a general form of ax + by = c, where "a" and "b" are numerical coefficients, "x" and "y" are variables and "c" is a numerical constant. Linear equations graph as straight lines, but graphing requires the equation be converted to slope-intercept form, which states y = mx + b, where "m" is the slope and "b" is the y-intercept. A system of linear equations is a set of two or more multivariable equations that can be solved at the same time because they're correlated.
Any straight line can be represented by an equation. This equation contains all of the information to then allow you to graph the line on a coordinate plane, consisting of x and y axes. The common form of the equation of a line is called the slope-intercept form, written as y = mx + b. The two coordinates, x and y, define where the line falls on the coordinate plane. You can easily convert any inequality into the slope intercept form, too.
Linear equations are simple equations that contain only variables and numbers. Linear equations graph as a straight line based on slope intercept form, y = mx + b where "m" is the slope of the line and "b" is the y-intercept, or point where the line crosses the y-axis. A system of linear equations is two or more linear equations that can be solved at the same time because of a correlation between the answers.
Linear equations graph as a straight line based on the slope intercept form, which states y = mx + b where "m" is the slope and "b" is the y-intercept, or point where the line crosses the y-axis. The slope creates the angle of the line and is defined by the distance between a point and the next point on the line. Slope is referred to as being "rise over run" because it causes a movement on the y-axis, then on the x-axis. For example, a slope of 2 would cause a point to shift 2 spaces up (y-axis) followed…
Equations with two variables -- "X" and "Y" -- are given as "a1X + b1Y = c1" and "a2X + b2Y = c2," where the letters "a1," "a2," "b1," "b2," "c1" and "c2" denote the numeric equation coefficients. The solution of this system is a pair of values ("X" and "Y") that simultaneously satisfy both equations. In mathematics, Cramer's rules allow you to easily solve such equations. The procedure is based on computing determinants for three equation coefficient matrices.
Linear equations describe straight lines or flat multidimensional surfaces. Systems of linear equations are sets of linear equations. They are found in many academic and technical disciplines. Linear equations are used in statistics, engineering, physics, finance, and economics. A given system of linear equations can fall into one of three categories. For the purposes of this article, the following two dimensional system will be used as an example: 4x + 5y = 1 4x - 2y = 2
Quadrilaterals are a family of four-sided, four-cornered polygons (or closed geometric shapes) that includes parallelograms, rectangles, rhombuses, trapezoids, kites and squares. All of the quadrilaterals have straight sides and interior angles that add up to 360 degrees. The shape names refer to particular figures, but each shape can belong to the category of other shapes, as well. For example, due to its rules, a square is also a rhombus.
A linear equation contains variables, or letters, representing unknown values, and constants, or numbers, combined with algebraic operations. When graphed, linear equations form straight lines. The purpose of a linear equation is to use algebra to isolate the variable on one side of the equation, thus solving for the variable and making all parts of the equation known. To properly solve an equation, the rules or properties of algebraic operations must be followed. The equality properties of addition and multiplication are two rules that commonly arise during the solution of a linear equation.
Whole numbers are 0 and the positive numbers to the right of it on the number line, starting with 1 and continuing with 2, 3, 4 and so on. Integers are the whole numbers as well as their negative equivalents, which extend to the left of 0, with -1 closest to 0 and -2, -3, -4 and so on moving farther and farther left.
The slope of a graph indicates the steepness of the curve on a chosen interval between two points. Slope is a simple measurement of a line and is a ratio of the difference between the y-values of two points on the line -- the "rise" -- divided by the difference between the x-values of two points on the line -- the "run." The larger the slope is, the steeper the line is between two points.
Quadratic equations are mathematical functions that take the form ax^2 + bx + c = 0, where a, b and c represent constant numbers and x is the function's independent variable. They describe the shape of parabolas, the speed of falling objects and the motion of pendulums. To solve a quadratic equation, find the values for x that result in zero. With practice, you can quickly factor some equations, such as x^2+ 2x -- 8, but not others, like x^2 + 2x - 9. For tougher cases like these, you solve using a method called "completing the square."
To graph a linear equation, which always produce straight lines, the equation needs to be converted to slope intercept form: y = mx + b, where "m" is the slope and "b" is the y-intercept. Both the "m" and "b" need to be known to put the equation in this form. If the "b" is unknown, but one point, point (x1, y1), is known, the point slope form can be used to get to slope intercept form: y - y1 = m(x - x1). The definition of slope involves the distance between points (x1, y1) and (x2, y2) and is…
Linear equations contain variables, or letter representations of unknown quantities, and numbers combined using algebraic operations. The general form of linear equations is ax + by = c where "a" and "b" are numerical coefficients, "x" and "y" are the variables and "c" is a constant. Linear equations graph as a straight line.
Numbers are easier to manipulate than things. If you're a cattle rancher, for example, it's easier to subtract 1,000 head of cattle from 1,500 head of cattle and know that leaves 500 cattle (1500 - 1000 = 500) than to go out on the range, herd and pen 1,500 cattle, separate 1,000 from the 1,500, and finally count the remainder. The manipulative characteristics of numbers are especially useful when you're simplifying equations, or isolating the variable in an equation. Isolating the variable in an equation is often your first and most important step when solving an equation.
Greek mathematician Archimedes (c. 287-212 B.C.) is famously quoted as saying, "Give me but one firm spot, and I will move the earth." Leverage is advantage. It's the ability to do a lot with a little. The slope and y-intercept of a mathematical line are leverage too. They are only two numbers, but if you know them, you can write an equation that will define the line and every point on it, whether the line is an inch or an infinity in length. Leverage your algebra. Learn how to determine a line's slope and y-intercept.
The slope-intercept form of a linear equation is a widespread way of displaying trends in the world. Even though it is used most frequently in algebra, it has many real-world applications. In fact, wherever there is a constant amount of change, a slope-intercept equation can help you understand that trend. This is relevant to trends in physics, economics, chemistry and many other situations. The slope corresponds to the change, and the y-intercept corresponds to the initial value. The slope-intercept form can be used when calculating the distance traveled by a jogger at a constant speed, for example.
Boolean algebra uses algebraic conventions to work with the logic values of true or false. The values 1 and 0 represent the two true or false values, and the operations of product, sum and negative represent the logical operations of "and," "or" and "not" respectively. Boolean algebra defines rules governing these values and their operations to devise a self-consistent mathematical system that is used in electrical controllers and computer systems.
"Every fool is a prophet in his own land." This is another way of saying there's always another side to every story or situation. It's often the same in mathematics. The expression (-1/3) x (-9/3), for example, is another way of writing the number 1. A point-slope equation and slope-intercept equation are two more ways of expressing the same thing. In this case, each defines a line in a two-dimensional Cartesian coordinate system, or x-y graph.
A matrix, in mathematics, is a rectangular array of terms commonly used in linear algebra to represent linear transformations. Matrix multiplication takes the product of a pair of matrices to create a new matrix. Matrix multiplication is not to be confused with scalar multiplication, which is the product of a matrix and a scalar value. This simply scales one matrix and does not involve producing a new matrix from the product of others.
Mathematicians often write the equation of a straight line in slope-intercept form. This equation looks like "y = mx + b", where x and y are the two variables being plotted on the line, m is the slope of the line and b is the y intercept. The slope is a measure of the angle of the line and is the amount the line goes up (its "rise") over a given horizontal distance (its "run"). The x and y intercepts are the points where the line crosses the x and y axes, and you can use these to generate the…
The equation that describes a straight line can have several different forms. The "standard" form has the general appearance of ax + by = c, where x and y are the two variables being plotted on the line, and a, b and c are numbers specific to each straight line. Mathematicians also describe a straight line in terms of its slope, m, as well as b, the point where it crosses the y axis -- the "y-intercept." The slope-intercept form of a straight line equation is y = mx + b. You can find the value of m and b…
The general form of a linear equation is ax + bx = c. But linear equations are solved via graphing, and forming a graph requires the conversion of the equation to slope intercept form, y = mx + b. In slope intercept form, "m" represents the slope, which expresses how many spots to the right and up a point is from the point before it. The "b" is the y-intercept, or the point at which the line intersects the y-axis.
A mathematical expression contains variables and numbers combined with algebraic operations. Inequalities represent the relationship between an expression and another expression or constant with an inequality symbol. The inequality symbols are > ("greater than"), < ("less than"), ≥ ("greater than or equal to") and ≤ ("less than or equal to"). For example, 2x > 8 is an example of a simple inequality. A compound inequality combines two or more simple inequalities. For example, 3 < 2x > 8.
Populations often grow exponentially rather than arithmetically. That is, they change by a constant ratio per unit of time rather than a constant addition per unit of time. For example, a population may double every year, rather than adding 10 units every year. The logarithm changes an exponential growth curve into linear growth, which can be easier to visualize. You can use Excel for some basic analysis in this realm, but sophisticated analysis depends on time series software.
Linear equations are simple equations with a general form of ax + bx = c where "a", "b" and "c" are given numbers with "a" and "b" serving as coefficients and "c" as a constant. Linear equations graph as straight lines in the rectangular coordinate system. In order to graph the line, the equation needs to be converted to slope-intercept form, which states that y = mx + b where "m" is the slope and "b" is the y-intercept, or point at which the line crosses the vertical axis.
Linear equations have a form that includes variables (unknown quantities represented by letters), coefficients (numbers attached to variables with multiplication) and constants (numbers) but no roots or exponents. Linear equations graph as a straight line on the rectangular coordinate system. The graphing form of a linear equation is slope intercept form, y = mx + b , where "m" is the slope and "b" is the slope intercept, or point where the line intersects the y-axis.
Linear inequalities contain inequality symbols that denote the relationship between the two segments. The symbols are > ("greater than"), < ("less than"), ≥ ("greater than or equal to") and ≤ ("less than or equal to"). Linear equations graph as straight lines based off the slope-intercept form of y = mx + b, where "m" is the slope of the line and "b" is the y-intercept, or the point where the line intersects the y-axis. Linear inequalities can use the same form, replacing the equal sign with an inequality symbol.
Linear equations can include constants (numbers), variables (letters representing unknown values) and coefficients (numbers multiplied by variables). The general form of a linear equation is ax + by = c, where "a" and "b" are coefficients, "x" and "y" are variables and "c" is a constant. Linear equations graph as a straight line. The general form isn't helpful in graphing. For graphing, convert the equation to slope-intercept form: y = mx + b, where "m" is the slope (replacing the "a" and "b" is the y-intercept, or point where the line crosses the y-axis.
Linear inequalities resemble linear equations in form except the equation's equal sign is replaced by an inequality symbol. Inequality symbols represent the size relationship between the two sides of the inequality. The symbols are > ("greater than"), < ("less than"), ≥ ("greater than or equal to") and ≤ ("less than or equal to"). In the latter two inequalities, the two sides have the potential of equaling the same value when solved. Thus the graphed lines of those equations include the line itself in the solution set.
Linear equations can contain variables (letter placeholders for unknown values), coefficients (numbers attached to variables through multiplication) and constants (numbers on their own) combined through algebraic operations. These equations won't include exponents or roots. A linear equation will graph as a straight line on a rectangular coordinate system.
The rectangular coordinate system used in all algebraic graphing is built around a cross-shaped set of axises. The vertical axis is the y-axis and the horizontal axis is the x-axis. The point at which the two meet is the origin and it carries a graphical point of (0, 0). The purpose of graphing is to provide solutions, and those solutions, or solution sets, are found by analyzing the relationship of the data to the known structure of the coordinate system.
LInear equations are a simple form of equation that contain only variables and numbers, which appear either as coefficients (multiplied to variables) or constants (alone). There can't be exponents or roots in linear equations. Solving a simple linear equation involves using algebra to isolate the variable on one side of the equation. Though most linear equations have one solution, it is also possible that the equation is false and thus incapable of producing a valid solution.
The Lorenz curve is a graphical representation of the degree of inequality in a sample from a population. It is often used in economics to measure inequality of wealth or income and in ecology to represent inequality of size of organisms. If there is perfect equality -- for example, if all individuals make the same amount of money -- the Lorenz curve is a straight line at 45 degrees. Any inequality shows as a gap between the Lorenz curve and a 45 degree line. If there is perfect inequality -- for example, if one person has all the wealth --…
Linear inequalities are similar to linear equations in that they can contain variables, coefficients (numbers attached to the variables) and constants (numbers on their own) but not exponents or roots. Linear inequalities include an inequality symbol in place of the equal sign. The symbols denote the size relationship of the two sides of the equation. The symbols are > ("greater than"), < ("less than") ≥ ("greater than or equal to") and ≤ ("less than or equal to").
Power is the rate at which energy is supplied to an object or converted to a different form. As an example, energy from an electrical circuit may be converted to a different form, such as mechanical energy, and power provides the rate at which this conversion takes place. The form that the equation for power takes depends on the form of energy that is being converted; however, the basic equation for power can be derived from the energy equation in a few steps.
Equations contain terms separated by an equals sign that shows they are equivalent. Equations have variables, or letter representations of unknown quantities, that need to be solved. Algebra is used to isolate the variable on one side of the equals sign, leaving the numerical solution on the other.
The general definition of a sentence in mathematics is that it states something involving numbers. The sentence can be all numbers, such as 5 = 5; include numbers and a variable, such as 5x = 15; or it can involve words, such as "3 is an odd number."
A system of equations is a set of multivariable equations that can be solved at the same time due to their correlation. When there are two equations in the system, with two variables each, the easiest way to solve the set is through substitution. This is done by using algebra to isolate a variable in one of the equations. The value of that variable is then plugged into the other equation to solve for the second variable. The solution is then used to solve for the isolated variable.
A function describes the relationship between two or more variables in mathematics. Often, there are just two variables (x and y), and these functions can be graphed on a two-dimensional grid made of horizontal and vertical lines. The x axis is the horizontal line at which the value of y is zero, and the y axis is the vertical line at which the value of x is zero. The point where the function crosses the y axis is its y intercept. You can readily calculate the y intercept by recognizing that the value of x will be zero at this…
A linear relationship occurs when one variable depends on the value of another variable. These relationships can be represented in a linear equation with letters taking the place of the unknown variables. Linear equations contain a variable(s); coefficient(s), or numbers attached to the variable(s) by multiplication due to correlation; and constants, or numbers with no variable attached. Solving a linear equation involves using algebra to isolate the variable on one side of the equal sign while a solution constant is on the other side.
Linear equations have a slope intercept form of y = mx + b, where "m" is the slope and "b" is the y-intercept. Linear inequalities are similar to linear equations except an inequality symbol replaces the equals sign. The inequality symbols are > ("greater than"), ≥ ("greater than or equal to"), < ("less than") and ≤ ("less than or equal to"). Both linear equations and linear inequalities graph as a straight line, though inequalities also involve shading to denote the inequality symbol.
Inequality symbols denote the size relationship between the segments they separate. The symbols represent greater than (>), less than (<), greater than or equal to (≥) or less than or equal to (≤). Linear inequalities operate, and graph, similarly to linear equations with the exception that an inequality symbol replaces the equal sign. This means that linear inequalities have their own version of the slope intercept form, which is y = mx + b where "m" is slope and "b" is the y-intercept.
Linear equations graph as a straight line and can be placed in slope intercept form, y = mx + b where "m" is the slope of the line and "b" is the y-intercept, or point where the line hits the y-axis. Linear inequalities also graph as straight lines and can be converted to a form similar to the slope intercept form except the equal sign is replaced by an inequality symbol. The inequality symbols, which denote the relationship between the segments, are > (greater than), < (less than), ≥ (greater than or equal to) and ≤ (less than or equal…
More than one line can share one particular coordinate plane in Algebra. In fact, it's very likely that two lines will intersect at a given point on the coordinate plane. When they do, both lines share the same point, and for the equations that represent both lines, the x and y variables will be the same numbers. This point is known in mathematics as the point of intersection.
A clay triangle is a piece of laboratory equipment used in the process of heating substances. It is used in conjunction with other lab equipment to create a stable framework in which to place a substance -- usually a solid chemical -- while it is heated to a high temperature.
Place a dot on a sheet of paper. Place a second dot anywhere you wish on the same sheet of paper. You have just created a line. Now, do the same thing on an x-y axis. This time, your line represents a graphed function. The numerical slope of the line will indicate how much the line slants up or down, and the y-intercept of the line will tell you where the line meets the y-axis of the graph. You can use this graphed function's slope and y-intercept to write its equation.
To factor a number, you must determine the unique prime factors of that particular number. The same concept is used when determining common factors for a polynomial. You are looking for simpler polynomials with integer coefficients that, when multiplied together, give you the polynomial you started with.
Although you can find x-intercepts by setting the equation equal to zero and solving for the "x" value, having a graphing calculator graph out the equation is a quick and efficient way to find all the x-intercepts in the equation. By knowing which buttons to push on your calculator and which functions to use, finding the x-intercept can take less than a minute.
Quadratic equations have the standard form ax^2 + bx + c = 0 where "a" and "b" are coefficients (numbers attached to variables) and "c" is a constant (number without a variable.) The squared exponent must be present for the equation to be quadratic, thus the name as "quad" means "square." Any equation that can be manipulated, with algebra, to fit this standard form is a quadratic equation. Quadratics graph as u-shapes called parabolas.
A graphed straight line is generated by a linear equation. Linear equations contain variables, coefficients (numbers attached to variables) and constants (numbers alone), but no roots or exponents. The most common form of linear equations is y = mx + b, where "m" is the slope and "b" is the y-intercept. The slope is what decides how much, and in what direction, the line slants.The y-intercept is the point at which the line intersects with the y-axis.
Solving a two step equation, or a multi-step equation, on a calculator can be tricky. First of all, the calculator can't know about the order of operations required to get the right result. Second, the calculator is programmed to operate in a particular way, and different calculators may use differing protocols. To avoid these headaches, the best thing to do is to set the order of your calculations yourself, then do the individual operations on your calculator. This sidesteps any mathematical order of operation problems that might occur, and uses the calculator for what it does best: crunching complex figures.
Calculators may be useful, but they're not always necessary. Newton and Leibniz developed calculus without the help of a calculator, so it should not be surprising that you can investigate the fundamentals of calculus and pre-calculus without a calculator, as well. The slope and y-intercept of a line are two such fundamentals. A line's slope and y-intercept identify that line by defining the specific function that creates the line. A simple equation known as the point-slope formula makes finding the slope and y-intercept of a line easy without a calculator.
Not all equations have a solution. Some have an infinite number of solutions. For example, trigonometry equations are periodic, which means their values repeat themselves over and over again. So, if a trigonometry equation has one solution, it has an infinite number of potential solutions. Which method you use to solve an equation with many solutions will depend on the type of equation you are dealing with. A common example of equations with an infinite number of solutions, which may appear even in basic math courses, is a system of linear equations.
When you write a long expression in an equation, you must specify the order of operations to ensure that the expression is evaluated correctly. For example, if asked to evaluate "two times x plus one," you could multiply two times x and then add one, or you could add one to x, and then multiply the result by two. Mathematicians refer to expressions in equations as "quantities," and they enclose them in parentheses and then in brackets, in that order. Work from the inside out when you evaluate a long expression, first evaluating quantities in parentheses, then those in brackets.
Linear equations contain variables (letters, such as "x" or "y", representing unknown values) with leading coefficients (numbers multiplied to the front of the variable) and constants (numbers without variables attached.) This type of equation does not contain exponents or square roots. Linear equations graph as a straight line, starting with a point on the y-axis and finding additional points using the slope, which is represented by "rise over run", or moving a certain number of points up then to the right.
Put strawberries into a blender and a smoothie comes out; put carrots into a blender and chopped carrots come out. A function is the same: it produces one output for each individual input and the same input cannot produce two different outputs. For example, you cannot put strawberries into a blender and get both a smoothie and chopped carrots. This is what mathematicians mean when they write a function such as f(x) = x + 1. Put strawberries (x) into the function, and you get a smoothie (x + 1).
Two-step equations contain variables which can be solved in exactly two steps. The methods used to solve these equations are the same methods used in one-step equations. The order in which the methods are utilized is important in solving two-step equations. In order to solve a two-step equation, you need to get the variable on one side of the equation and a numerical value on the other side of the equation using two operations.
In order to solve an equation, whether it is a simple one-step equation or a multi-step equation that requires lengthy figuring and calculation, you need to isolate the variable or variables on one side of the equals sign and the numbers on the other. Multi-step equations involve basic addition, subtraction, multiplication and division, along with more complicated operations, such as squaring.
If a farmer gave you a flock of sheep and said, "Keep warm with this," you might ask "How?" However, if that farmer sheared the sheep, carded and spun the sheep wool into yarn, and then knitted you a sweater, you'd know exactly how to keep warm with the sweater. Math equations are often the same: in one form unusable or solvable, but in another form, easily solvable. Completing the square, or making a quadratic expression into a perfect square, is one way of solving an equation.
Linear inequalities are similar to linear equations except the equal sign of equations is replaced with an inequality symbol. The inequality symbol denotes the relationship of the segments it separates. The symbols are > ("greater than"), < ("less than"), ≥ ("greater than or equal to") and ≤ ("less than or equal to"). Like linear equations, linear inequalities graph as straight lines but they include shaded-in areas that indicate the inequality symbol.
Every day you manipulate objects around you to make them more useful. You pour hot coffee in a cup before drinking it; you fry an egg before eating it; you turn up the volume of your iPod in order to hear it on a crowded street. Mathematicians do the same thing with equations and inequalities; they manipulate them to make them useful, that is, to make the equations or inequalities solvable. The method of elimination, or the addition method, is one type of mathematical manipulation.
Scientific investigation is the testing of a hypothesis by carrying out experimental data collection. The data is often tabulated into a table format, but this does not allow trends to be identified easily. A graph is a visual representation of data, and there are many different types, depending upon the data being plotted.The most common form involves plotting points on horizontal and vertical axes. This is known as a scatter plot and is commonly used to identify the mathematical relationship between two variables.
Linear equations graph as straight lines. The general form of a linear equation is ax + by + c = 0, where "a" and "b" are coefficients of the variables "x" and "y." The slope intercept form is y = mx + b, where "m" is the slope of the line and "b" is the y-intercept (point where the line intersects the y-axis.) The point slope form is y - y1 = m(x - x1) where (x1, y1) references a specific point on the line.
A graph is a visual way of representing data, and there are several different types that can be used. An x-y graph plots data points according to their value on a horizontal x-axis and vertical y-axis. Once plotted, the points are joined to form a curve or straight line. A common technique for analyzing data is to check the slope (also known as gradient) of the curve/line The gradient shows the rate of change of the "y" variable with respect to the "x" variable.
Unlike equations that only have one or two single answers, inequalities concern problems that have an entire set of numbers in them. Inequalities can come in a range of forms, from linear forms with one symbol, to ones with three sections in them. In this case, the goal is to find a set of numbers that sit in between the two outer sections.
A line is a straight curve that extends infinitely in both directions. A line segment is a portion of a line that is bounded at both ends. A line is defined by the equation y = mx + b, where m is the line's slope, or the rate at which it rises in the y-direction for each unit in the x-direction, and b is the y-intercept, which is the x-coordinate at which the line intersects the y-axis. If you draw a line segment on a sheet of graph paper, you can measure the endpoints (x1,y1) and (x2,y2) and use that…
Most people are used to linear equalities such as 2x + 3 > 5. Unlike a linear inequality, a quadratic inequality such as x^2 + x + 5 > 7 has two values in which a solution can fall in between. Only this set of numbers will work to solve the inequality; no other number will. Also unlike a linear inequality, however, to solve the inequality, you have to solve a corresponding equation that makes up the inequality.
The rate of a chemical reaction is expressed in terms of how rapidly reactants change into products. Each reaction has its own rate law, and this rate law can only be found from experimental data; you can't find it by chalkboard calculations alone. Nonetheless, once you know the rate law for a reaction, you can predict how a change in temperature will change the rate constant and in turn the rate of the reaction.
A system of equations is a set of two equations that share two variables, typically x and y. While both equations typically have numerous values that would solve the variable, the goal is to find the one set of x and y coordinates where the two lines intersect, as these numbers will solve both of the equations. To do this, you must graph both of the equations on the same coordinate plane.
A core of high school mathematics is algebra class, where students are taught to solve equations made up of letters representing numbers. In a typical algebraic equation, a dependent variable will be a function of a number of quantities. For example, the area of a rectangle can be calculated by multiplying the length by the width. This can be written algebraically as x = wl , where "x" is area, "w" is width and "l" is the length.
Matrices are sets of information stored within a bracketed system. The number of rows and columns defines the size of a matrix. For example, a 3 x 2 matrix would have three rows and two columns. A 2 x 2 matrix is called a square matrix due to the matching row and column numbers. Entering a matrix into a scientific calculator allows the user to use the calculator to perform algebraic operations and transformations on the matrix.
Graphing the intercepts and slope of a function f(x) involves finding its derivative function f '(x). This involves using either the definition of the derivative: (limit as h approaches 0) f(x + h) - f(x) / h; or using the basic rules of differentiation such as the power rule, quotient rule, multiplication rule and chain rule. Knowledge of these basic principles of calculus are necessary to complete this task. The key is to understand that the value of f '(x) at a given point is the value of the slope of the graph of f(x) at that point. The x-intercepts…
If you were given the equation x + 2 = 4, it probably wouldn't take you long to figure out that x = 2. No other number will substitute for x and make that a true statement. If the equation were x^2 + 2 = 4, you would have two answers √2 and -√2. But if you were given the inequality x + 2 < 4, there are an infinite number of solutions. To describe this infinite set of solutions, you would use interval notation, and provide the boundaries of the range of numbers constituting a solution to this inequality.
Drivers' inability to control traffic flow undoubtedly causes frustration and impatience as they sit behind other vehicles at intersections, construction zones or accident sites. While drivers can't fix real-life traffic woes, they can direct virtual traffic to ease some of their frustrations. Online traffic intersection games challenge players to control traffic lights and ensure that both vehicles and pedestrians travel safely.
A system of equations contains more than one equation with multiple variables that correlate so that the equations can be solved simultaneously. Methods of solving systems include substitution, where you solve one of the equations for a variable then you plug that equation into the first equation in place of said variable and solve; elimination, where you modify the equations to add and eliminate a variable to solve for the other variable, then you plug the answer back into the original equation to solve; and Gaussian elimination, in which you simplify systems where there are more variables than equations.
Linear inequalities are similar to linear equations except an inequality symbol replaces the equal sign of the equation. Inequality symbols are placed between portions of an expression to represent their relationship. The symbols are > ("greater than"), < ("less than"), ≥ ("greater than or equal to") and ≤ ("less than or equal to".) Solve an inequality the same way an equation is solved with one exception. If you multiply or divide across the symbol by a negative number, the direction of the symbol changes.
Linear inequalities look nearly identical to linear equations with the exception that an inequality symbol replaces the equals sign. The inequality symbols are > ("greater than), < ("less than), ≤ ("less than or equal to") and ≥ ("greater than or equal to.") An example inequality a + x < y means "the solution of a plus x is less than y". The inequality can solve for a variable like an equation and the answer can be represented graphically on a number line.
A ratio expresses the relative quantities of two different things. Math students will have to solve a lot of word problems involving ratios. The math is the easy part. Figuring out how to translate the word problem into the proper mathematical language takes some practice.
Functions are mathematical expressions where the value of one variable will turn up only one correlated variable answer. For example, in an expression containing the variables "x" and "y", the solution of "x" will only lead to one "y" answer. Functions have a form similar to F(x) = x^2 + b. To rewrite a standard equation in function form, isolate the "y" on one side and replace it with the "F(x)" symbol, as the two are equivalent.
Inequalities are similar to equations with the exception that the equals sign of equations is replaced by an inequality sign such as "<," which means "less than," or ">," which means "greater than." Inequalities can therefore be solved algebraically the same ways that equations can with one slight change. If a negative number is multiplied or divided across the inequality symbol while solving, that changes the direction of the symbol. The < sign would change to a > sign, for example.
A semicircle is the top or bottom half of a circle. They can be expressed using the equations "y=(r^2-x^2)^(1/2)" for a semicircle whose diameter is parallel to the x axis and "x=(r^2-y^2)^(1/2)" for a semicircle whose diameter is parallel to the y axis. In these equations, "R" is the radius of the semicircle. A semicircle whose diameter is parallel to the x axis can have one y intercept and two x intercepts while a semicircle whose diameter is parallel to the y axis can have one x intercept and two y intercepts. These intercepts can be found using simple mathematical…
A linear equation typically contain two variables and is in the form of y=mx+b. Each variable is called an unknown. Linear equations can also contain more unknowns than just two. For example, 2x+4y+5z-6=0 is a linear equation of three variables. A linear system of equations is a set of linear equations. The solution to a linear system of equations can be found by first representing the linear system in matrix form and using the Gaussian elimination method to find its solutions.
When graphed, linear equations form straight lines. These equations can be written in slope-intercept form for easier graphing by hand. The slope-intercept form is y = mx + b where "m" is the slope of the line, "b" is the y-intercept, and "x" and "y" are points on the graph. The slope of a line is represented by "rise over run" or movement up followed by movement to the right (reversing directions if the number is negative).
The absolute value of a number determines how far that number is from 0 without concern to whether it is in a positive or negative direction. Thus numbers within an absolute value symbol (a vertical line on each side) always result in a positive. Linear inequalities are similar to linear equations except an inequality sign replaces the equal sign. Linear inequalities where the absolute value of x is less than "a" follow the form -a < x < a. If a is greater (<a), the solution takes two forms: x < -a and x > a.
The zeros of a polynomial equation are the values of x that make the polynomial equal to zero. These values are referred to as roots. Depending on the complexity of the polynomial, determining roots by hand can be time consuming and make you prone to err. The TI-89 line of graphing calculators can solve many types of these equations, including trigonometric, rational and high-degree polynomials. Knowing the process for finding polynomial roots on your calculator can save you lots of time and ensure a correct answer if the necessary data is entered correctly.
Linear equations with multiple variables can be placed into a system for solving when the variables are co-dependent. The easiest to solve systems are already in row echelon form, where one of the variables is set equal to a number that can then be substituted back through the other equations. But more complicated systems, such as those having more variables than equations, can be solved using Gaussian elimination. Gaussian elimination can involve interchanging equations and multiplying one equation by another to replace the latter equation.
The point-slope form applies to linear -- or straight line -- equations, for which you are given the line's slope and a specific point (x1, y1) on the line. The form states that y - y1 = m(x - x1), where "m" is the slope and "x" and "y" are random points on the line. Equations in point-slope form can be rearranged into slope-intercept form (y = mx + b, where "b" is the y-intercept) for easier graphing.
Inequalities are equations in which two segments are compared using an inequality symbol. The symbol > means that the preceding segment is greater than the segment after it. The symbol < means that the preceding segment is less than the following segment. Inequalities can be solved as if the symbol were an equal sign with one exception. If a negative number is multiplied or divided across the symbol, that changes its direction. An < would become a > and vice versa.
Linear equations with multiple, related variables can be solved using a system of linear equations. Such systems can be simplified using row echelon form, which places a variable equal to a constant (such as z = 2) at the bottom of the system to allow for easy back substitution into the other equations. Complex systems can be simplified into row echelon form using Gaussian elimination. Gaussian elimination allows you to switch the places of equations and to multiply an equation by a nonzero number, then adding it to another equation to replace that second equation.
When you begin taking algebra courses, one of the things you will be taught is the concept of graphing an equation using a coordinate plane. The coordinate plane can be used when dealing with equations that may have more than one answer; the answers lie on the graph itself. Before you can start dealing with these equations, the best thing to do is identify whether or not they are graphing equations in the first place.
The Cartesian Plane is the standard cross-shaped graphing plane taught to children in school. The vertical line is called the "y-axis" and the horizontal line the "x-axis." The point at which they meet (0, 0) is called the origin. Traveling up from the origin on the y-axis yields positive numbers and down negative. Going right on the x-axis from the origin is positive and left is negative. A graphed point will have an x value listed first, then a y value.
At first glance, nature is bedazzling, what with buds opening, birds soaring, earthworms crawling and deer darting into forests---a spectacular confusion of living things, great and small, continually in motion. Using mathematics, biologists can reduce the confusion, unmasking patterns common to species from microbes to whales. One such pattern, underlying population growth, appears in the sigmoid curve.
Linear equations create straight lines when graphed. Graphing linear equations is made easy when they are converted to the slope-intercept form. The slope-intercept form states that y = mx + b where "m" is the slope and "b" is the y-intercept, or the point where the line hits the y-axis of the graph. The form can also be used to find the x-intercept, or point where the graph hits the x-axis, by setting "y" equal to zero and solving for "x."
Linear systems are sets of linear equations with multiple variables that can be solved together due to an interrelation. Linear systems already in row echelon form are easier to solve through back substitution of a known variable. More complex linear systems can be solved using Gaussian elimination, which employs the operations of switching equations, multiplying an equation by a nonzero number and then adding it to another equation to replace the latter equation.
Inequalities are mathematical expressions comparing two values according to which are largest or smallest. Inequalities contain the symbol "<," which means "less than," or ">," which means "greater than." Inequalities can be solved for an exponent the same way linear equations are, by using arithmetic to isolate the variable on one side. The main difference in solving an inequality rather than a linear equation is that dividing both sides of the inequality by a negative number requires flipping the inequality sign in the opposite direction.
Graphs consist of values on a horizontal x-axis and a vertical y-axis. Intercepts are the points at which a graphed entity crosses one of the axes. An x-intercept crosses the x-axis and a y-intercept crosses the y-axis. Linear equations, or those that create straight lines, can be rewritten into the slope-intercept form of y = mx + b, where "b" is the y-intercept. The x-intercept, in linear and nonlinear equations, can be found by setting "y" equal to 0 and solving for "x." The y-intercept in nonlinear equations is found by setting "x" to 0 and solving for "y."
Mathematical equations can be composed of variables, or letters (such as x or y) representing unknown quantities; exponents or roots (such as 2^2 or √4); and constants (such as 3 or 1/4). Linear equations are those that, when graphed, will create a straight line, usually at an angle or slope. Nonlinear equations can take a variety of curved shapes when graphed.
Matrices are bracketed sets of numbers that can be worked on algebraically. The size of a matrix is denoted by its number of rows by the numbers of columns, such as 3 x 3. Two matrices can be multiplied only if the number of columns of the first matrix equal the number of rows in the second matrix. For example, a 2 x 3 matrix can be multiplied by a 3 x 2 matrix.
Linear equations represent a constant relationship between variables. x = 2y + 2 is an example of linear equation. An increase in x results in a steadily accelerating increase in y. Equations of this sort will produce straight lines when plotted onto a graph and have a huge number of real-world applications.
You can readily find the equation for a straight line from the slope of that line and its y-intercept. This is because these two numbers essentially define any straight line. The slope is a measure of the angle of the line, and the y-intercept gives you the exact location of the line in two-dimensional space, so any particular combination of slope and intercept describes a single, unique line. You can readily write the equation of a line from its slope and y-intercept since these are the only two numerical values in the equation.
Mathematics is a fundamental subject, and is learned from an early age. Although math starts with the study of numbers, later on algebra deals with the representation of numbers with letters and symbols. Equations are algebraic problems, and they are typically solved analytically using a set of rules. Another way to solve equations is by plotting both sides of the equation on a graph in order to see where they are equal.
In mathematics, sometimes you are asked to work out the x and y coordinates for a point or line drawn on a graph. Usually these questions show a pair of numbered x-and y-axes. The x-axis is the horizontal line, and the y-axis is the vertical line. The numbers on the axes need to be equally spaced the axes should meet at zero. Even when there are few numbers labelled on the axes, finding the x and y coordinates is relatively easy.
Logistic growth is a form of population growth first described by Pierre Verhulst in 1845. It can be illustrated by a graph that has time on the horizontal, or "x" axis, and population on the vertical, or "y" axis. The exact shape of the curve depends on the carrying capacity and the maximum rate of growth, but all logistic growth models are s-shaped.
In algebra, a polynomial or quadratic equation is an equation of the form ax^2+ bx+c=0, where a, b and c are integers. Because it is an equation, it can be solved in terms of x, giving two distinct values for x. In many cases, the easiest way to solve for x is to use the quadratic formula, x=(-b±√(b^2-4ac))/2a.
The x-intercept and y-intercept of a graph are the points at which a line crosses the x or y axis. On a graph, the x-axis is the line running horizontally while the y-axis runs vertically. When dealing with the equation of a line, the x-intercept is where the y variable equals 0 to represent hitting the axis. Likewise, the y-intercept is where the x variable equals 0. The Slope Intercept Form can help solve linear equations. The formula is y = mx + b, where "x" and "y" represent graphical points, "b" is a y-intercept (point where the line hits…
Data representation in science can take many forms, from tables to graphs. In an experiment, you will have an independent variable (y) and a variable that is measured, called the dependent variable (x). A common way to visualize this data is via a line graph whereby points are plotted based upon the x and y values. Another interesting way to identify a data trend is by looking at the rate of change of the dependent variable with the independent variable. This is known as the gradient or slope.
Finding the x- and y-intercepts of an equation are important skills you'll need in math and the sciences. For some problems, this may be more complicated; fortunately, for linear equations it just couldn't be simpler. A linear equation will only ever have, at most, one x-intercept and one y-intercept.
The multiplication of matrices requires that the second matrix have an equal number of columns as the first matrix has rows. For example, you can multiply a 2 x 3 matrix by a 3 x 2 matrix, but you can't multiply a 2 x 3 matrix by a 5 x 3 matrix. There is a formula for solving matrix multiplication, but it is more complicated than simply doing out the multiplications by hand; the "by hand" method is especially recommended for those new to doing matrix multiplications in order to reinforce the concepts applied.
You can find the equation of a straight line as long as you know any two points on that line. Because the x intercept is the point where the line passes through the horizontal axis, and the y intercept is the point where it passes through the vertical axis, you can use those two points. If you use these two points to calculate the slope of the line, m, you can can write an equation for the line in the form of y = m(x) + b, where b is the y intercept, which you already know.
A matrix is a rectangular array of numbers. One matrix can be subtracted from another if it is of the same order -- that is, if it has the same number of rows and columns. Matrices are often used in statistics. Matrices are usually written surrounded by braces. Excel, the spreadsheet application included with Microsoft Office, refers to a matrix as an "array." It has built-in functionality to make working with matrices, or arrays, simple.
Slope and y-intercept are two of the most important elements of graphing lines. Typically, these elements are found in slope-intercept form linear equations, but can also be found in slope-intercept form inequalities. These inequalities vary from linear equations only in the fact that their solution is constant of either the left or right side of the line. Finding the slope and y-intercept of an inequality involves the same process as finding the slope and y-intercept in a standard linear equation.
Slope is the rate of change over the course of a line. In standard linear equations this is a constant rate, but in parabola, or curved equations this rate is variable at each point along the curve. Determining the rate of change in a curve is relatively the same as finding slope in a standard linear equation. To points are needed to find the slope at any particular point in a curve. The closer these points are the more accurate your slope will be.
Matrices, the plural form of matrix, are organizational mathematical data sets placed inside brackets. A matrix contains horizontal rows and vertical columns that designate its size. A 2 x 3 matrix, for example, has two rows and three columns. Matrices can be multiplied if the number of columns in the first matrix equals the rows of the second. You can, for example, multiply a 2 x 3 matrix by a 3 x 1 matrix. But you can't multiply a 2 x 5 matrix by a 3 x 1 matrix.
Mathematical word problems typically require a person to identify a specific equation from the text. These equations will usually contain any number of constants and variables that need to be properly arranged to solve the problem presented in the text. Writing these equations out in their properly arranged order requires a minor amount of contextual identification within the word problem, and can be done by virtually anybody.
A quadratic equation is a special kind of algebraic equation that has two or three terms on the left hand side. One of the terms contains the variable that you are solving for raised to the first power. A second term contains that variable raised to the second power. The third term, if present, does not contain that variable at all. The right hand side of the equation is zero. You can often solve quadratic equations by factoring them into their component terms.
Any line can be represented by a linear equation in the slope-intercept form y = mx + b, where m is the slope of the line and b is the y-intercept. The slope m is the steepness of the line and is determined by dividing a change in the y-coordinate value by the corresponding change in the x-coordinate value. The y-intercept is the y value where the line crosses the y-axis. The equation of a line, in y = mx + b form, can be found if you know the slope m and the coordinates of one point on the…
With the use of a matrix, mathematicians and scientists can input and manipulate data to find different outputs in an equation. The determinant, represented by the scale factor of a specific matrix, is the result generated from the data. Determinants can also be used to tell if a matrix can be inverted as a singular matrix.
A sigma math equation simply refers to an equation using sigma notation. Sigma notation is the Greek letter "sigma" --- which looks a bit like the capital letter "E," with an n on top and an x = (an integer) on the bottom --- and is called a Riemann sum. When this notation precedes an equation, it indicates to solve the equation by finding the sum of the equation between x and n. The x is the lower bound of the equation and n is the upper bound of the equation. To solve the Riemann sum of (4 + 5x)…
Unlike an equation, an inequality refers to a problem in which the solution is a group of real numbers, rather than just one real number. The basic forms of inequalities are mathematical statements such as "2 < 3" and "3.467 > 3.466." More complex versions of inequalities involve variables; by graphing these inequalities, you can visualize their solutions.
Linear difference equations are mathematical expressions that are defined by previous solutions of the equation. Because of this, they are sometimes called recurrence equations. For example, a specific linear difference equation is a sequence of terms. The sixth term of the sequence depends on knowing the fifth sequence of the equation, which depends on the fourth sequence and continues back until the initial term. These types of equations are frequently used to numerically solve linear differential equations, as the discrete version of a differential equation is the difference equation. Linear difference equations generally can be solved in a series of…
Released in 2004, the TI-89 Titanium is the latest addition to the TI-89 series of calculators. It is a scientific graphing calculator that unlike other Texas Instruments calculators, can give answers that include single or multiple variables. The TI-89 Titanium can also solve equations with respect to variables. This can be very useful with particularly difficult equations or when solving equations outright is impractical. Solving equations is a simple task done by manipulating the TI-89 Titanium's solve feature.
Solving a system of linear equations can be done by hand, but it is a task that is time-consuming and error-prone. The TI-84 graphing calculator is capable of the same task, if described as a matrix equation. You will set up this system of equations as a matrix A, multiplied by a vector of the unknowns, equated to a vector B of constants. Then the calculator can invert the matrix A and multiply A inverse and B to return the unknowns in the equations.
Two circles can intersect in two points, be tangent and intersect at exactly one point, or not intersect at all. Given the origin of each circle, and its radius, you will first need to determine if the two circles don't intersect -- that is, whether they are too far apart or if one is contained completely within the other. Then you can check if they are the same circle, in which case they have infinitely many intersections. If the two circles do intersect at one or two points, you can pinpoint which two points.
In basic physics, it is often necessary to deal with linear motion. Linear motion is defined as motion along a straight line. It is therefore easily calculable using basic algebraic equations. There are four basic linear motion equations that deal with five variables. Each of the equations contains four variables, meaning you need to know at least three variables in order to solve the equations. The five variables in these equations are as follows: "a" is the acceleration, "u" is the initial velocity, "v" is the final velocity, "t" is the elapsed time and "s" is the distance.
Imaginary numbers and complex numbers (numbers with both real and imaginary components) are common in mathematics and electrical engineering. While the TI-84 Plus will readily display imaginary numbers, it is less obvious how to enter imaginary numbers. It supports two methods of entering complex numbers: rectangular mode and polar mode. In rectangular mode, you will enter the number as the sum of a real part and a complex part. In polar mode, you will enter the number as a radius and an angle.
Inequalities crop up frequently in math and science. You may be familiar with plotting these on a number line using open and closed circles to represent strict inequality and "or-equal-to" inequalities, or you may be new to plotting inequalities on a number line altogether. No matter the convention used (circle or brackets and parentheses), it displays a set of values a variable can take on while obeying an inequality.
The slope of a line is the ratio given by the change in the line's y (vertical) value divided by the change in the line's x (horizontal) value. The slope effectively describes the steepness of the line. Graph equations for which the slope value must be solved are common for middle school mathematics courses. The slope value can also be used to graph the line if either the x or y values are undetermined in the given equation.
Hexagons are a common shape in nature, from the honeycombs of bees to the sections of a tortoise shell. Many hydrocarbon groups are hexagonal and certain minerals like basalt have hexagonal crystal structures. Determining the area of a hexagon from its vertices can take a little work, though a regular hexagon has a simple area formula.
In chemistry, each reaction has its own rate law, a law that tells you how fast the reaction happens. Unlike chemical equilibria, rate laws cannot be predicted theoretically; they can only be determined experimentally. The order of a reaction tells you what exponent to use for a given reactant when writing a reaction's rate law. If you have already conducted an experiment and have experimental data you can use -- or if you are working a problem on a chemistry exam -- you can use this information to determine the rate law.
Though triangles are often specified by the coordinates of their three vertices, it is a format that may not seem particularly conducive to determining its area. Two methods can be used to determine a triangle's area given just its three points. One method involves less computation on your part but requires a 3-by-3 matrix determinant. The other takes a little more work, but does not involve a matrix determinant.
The logistic equation is an important mathematical model first developed by Pierre Verhulst in 1847. The logistic equation is commonly used as a model of population growth for a single species in an environment of limited resources. Therefore, the logistic equation finds important uses in many biological applications, such as growth of bacteria in a petri dish. While commonly written as a differential equation, the logistic equation can be solved to give an equation whose parameters can be changed to give a clear understanding into how the logistic equation should be used.
The term "linear equation" in this context refers to the differential model for population saying that the rate of growth of a population is proportional to the size of the population. The closed-form solution of this equation is exponential and so this linear model describes exponential population growth. Although sometimes more complex models are required to describe population growth, using two data points, this model can provide a good fit for many situations.
Converting between Fahrenheit and Celsius temperatures can give you fits if you are not accustomed to switching between the two. Both systems of measuring temperature are linear, and they cross at a single intersection point, meaning there is a single value that, when converted, will be the same in both systems. This information is helpful, for example, when visiting foreign countries so that you can interpret the weather forecast.
A physics problem may require you to determine the height of a dropped ball. In order to solve these types of equations, you must be aware of at least one variable. While there are four kinematic equations available, one does not involve the "distance" variable. This leaves you with three basic kinematic equations that can solve this problem. For the purpose of these physics questions, air resistance and other factors are generally disregarded.
When working with physics equations, you are sometimes required to find information about initial velocity, final velocity, acceleration and distance without knowing the amount of time that has surpassed. Luckily, by combining two basic kinematics equations, it is possible to make an equation that doesn't require the "time" variable to be present. This equation is v^2 = u^2 + 2(a)(s).
Solving linear equations that have denominators, the bottom part of a fraction, is not that different from solving those without denominators. Once you get rid of the denominators you can proceed with solving the linear equation like you would any other. The denominators can become visually confusing, however, with the possibility of variables and exponents showing up. These are solvable, however, so don't panic.
Linear equations are equations that have a single variable that acts on a function. The idea is to isolate each side so that your unknown variable is on one side, while the solution is on the other side. When you have multiple equations in a system and their slopes are not parallel, the system is known as a consistent system and the solution to the system is where the two lines cross on a graph. By plotting points on a chart along two axes, you can determine where a system of lines crosses.
Linear equations describe straight lines. They do not use squared, cubed or other exponent values except 1. A common homework problem involves solving for x and y in a linear equation, but it is also possible to take the points from a graphed line and work back to find the equation. A simple form of linear equation is called the slope-intercept form, or y = mx + b.
An algebraic equation consists of two variables. Mathematicians usually call these variables x and y. For example, in the equation, 2x + 3y = 12, the variables are x and y. Each variable has a coefficient preceding it. In this example, those coefficients are two and three. A system of equations consists of multiple equations. Solving a system of equations requires you to determine values for x and y that satisfy all equations. You can do this using a process called elimination.
A line equation, ax + by = c, has three different parameters: a, b and c. By changing the different parameters, it is possible to change the y-intercept, x-intercept and the slope of the line. The ability to manipulate a linear equation makes it possible to use lines in many ways. The most simple linear equations are vertical and horizontal lines, as these lines have no slope. The equation for a vertical line is x = a and the equation for a horizontal line is y = b. Other more complicated lines are not horizontal or vertical and will have…
Algebraic equations --- mathematical expressions that use undefined variables --- are often used to describe curves as they appear on a graph. Some well-known curves like circles or parabolas have specific equations associated with them, but you can create an algebraic equation to describe almost any curve.
When two non-parallel lines cross, they create an angle between them. If the lines are perpendicular, they form a 90-degree angle. Otherwise, they create an acute, obtuse or other type of angle. Every angle has a "slope." For instance, a ladder against a wall has a slope whose value varies according to the angle of the ladder. Using a little geometry, you can compute the angle between two intersecting lines by determining their slopes.
Absolute value is a concept divorced from the normal logic of the number line. Instead of indicating a specific number, an absolute value specifies distance of a number from zero. This means that the absolute value of 5 (symbolized as |5|) is both 5 and -5, which are both 5 steps away from zero on the number line. Absolute value can also be used in algebraic equations, meaning that an unknown value can be included in the equation and then solved for through algebra.
Isaac Newton's laws of motion changed human perception of the universe, from one in which an unfathomable heavenly sphere ruled over the Earthly world to a place governed by the same universal laws everywhere. As described, for example, in George Smith's essay "Newton's Philosophiae Naturalis Principia Mathematica," Newton's laws generally collected and codified principles that were already known -- but they were insights based on observation and logic, and were not "derived" in the usual sense of the word. However, following examples such as that given by Tom Kirchner in his notes for York University's classical mechanics class, it is…
You can calculate the area of a field, a state or even a country with the help of a GPS device. GPS stands for Global Positioning System. It is a navigational system that uses satellites, computers and precise clocks to determine the latitude and longitude of a receiver by triangulating the time differences of signals from at least three satellites to reach the GPS receiver.
You can calculate simultaneous equations on your TI-84 graphing calculator. Creating a program that solves simultaneous equations will require you place the coefficients of each equation into a matrix. The coefficients are the numbers representing the multiplier of each variable. For example, if the equation is "6x +7y = 13", the coefficients would be "6" "7" and "13".
A linear equation is an algebraic relationship in which the dependent variable appears raised to the power of one, meaning there are no higher-order exponential variables present. A linear equation of the multi-step variety will require two or more mathematical operations to fully solve for the value of the dependent variable. An efficient execution of such a solution requires following a specific order of operations. The general procedure for this is typically encountered in an introductory algebra course.
You may have heard the term ski slope. A ski slope is part of a mountain that has an incline. Skiers move down the slope at varying speeds. The steeper the slope, the faster the skiers move. In mathematics, slope refers to the inclination of a line. Lines consist of points called x and y coordinates. By examining the values of these coordinates, you can compute the slope factor of a line.
A common technique used to solve a system of coupled linear differential equations involves decoupling the equations through matrix methods and integrating each one separately. The key to the success of this method is the ability to decouple the equations by diagonalizing the square matrix which results when these equations are rewritten in matrix form. This technique requires matrix algebra, including eigenvalues and eigenvectors, and integral calculus, and there are many resources available on these topics such as Wolfram's Math World.
A linear equation is an equation in which there are no squares, cubes or higher powers. A set of linear equations can include two or more variables, but if there are more than two variables, it is very hard to graph them, and if there are more than three variables it is impossible, as each variable requires a dimension. Graphing a linear equation gives a graphical feel for how the equation works. Graphing two or more equations can also show if they are compatible--that is, if there are any values that satisfy all the equations.
Many people find math intimidating, and in particular, word problems. Solving word problems with linear equations, problems with more than one unknown variable, seems almost impossible to some people. No matter your age or profession, math and problem solving is everywhere, and you're bound to come face to face with the challenge of solving word problems at some point. Translating word problems from English to actual mathematical equations is the prevalent stumbling block most people face. However, once you construct the equation, solving for the answer is relatively straightforward.
Linear equations are an algebraic method that solves problems limited to the first power. The variables in the expression cannot be squared, multiplied and divided, or be a square root value. As of 2010, these equations are still part of math curriculums for algebra and beyond. In order to effectively study, tools such as computer flash games are available to give students, parents, teachers and the curious the ability to learn and practice.
A matrix is a grouping of coefficient arrays in which the elements of one row combine to form a single equation, and the elements of one column denote a group of coefficients in the same variable. The linear matrix grouping of equations is something of a "short hand" for writing out systems of equations that abide by the same mathematical rules. Solving systems of equations with matrices is not a difficult task, but it can be time consuming if it is done by hand for a system of many variables.
A quadratic linear equation is an algebraic relationship that contains at least one second order term of the dependent variable along with one linear term of the dependent variable. A second order term is one which is raised to the power of two. A linear term is one which is not raised to any power (or is raised to the power of one). The common form of this type of equation appears as: ax^2 + bx + c = 0. Not all quadratic equations are initially given in this form, but they can be re-arranged to fit the common form.…
A system of linear equations includes two or more equations of two or more variables, which are all raised to the power of 1. In order to solve a system of linear equations, you must have at least one equation for each unknown variable. Matrices allow you to solve systems of equations more quickly, particularly when there are more than two unknowns.
Many people have seen the most basic of linear equations, y = mx + b. From this, a whole branch of mathematics has formed through the centuries. Linear equations, or solutions based on linear equations, are around us every day--from the cars we drive, to the computers we use, to the buildings we occupy. Without linear equations, we would lead much simpler lives.
The slope-intercept form of a linear equation is y=mx+b. m is the slope of the line that the linear equation represents. "B" is the height, or y-value, where the line intersects the x-axis. This form is not be confused with the slope form, which is m=(y-y0)/(x-x0), with (x0,y0) being any specified point on the line.
A specific algebraic formula we can use to understand the world around us is the linear equation. Sometimes when we want to understand something we have to account for all the variables involved. When you see a horse galloping around a race track and you wonder just how fast she is running, you can use the linear equation: s = d/t (speed is equal to distance divided by time) to discover how fast she was running.
Chemists use rate constants to show how the rates of chemical reactions are influenced by the concentration of a reactant. There are three primary types of reactions in terms of how reaction rate is affected by reactant concentration: zero-order, first-order and second-order. In zero-order reactions, reaction rates are constant regardless of reactant concentration. In first-order reactions, reactions rates are proportional to the reactant concentration. In second-order reactions, reactions rates are proportional to the square of the reactant concentration. In each case, calculating rate constant is relatively simple.
You can use the Gaussian method to solve a system of three linear equations simultaneously, if the system has a solution. The basic idea is to add integer multiples of two equations to produce a new equation with fewer variables. The new equation will substitute for one of the two equations from which you calculated it. You'd then repeat this process until all variables are determined. This approach is also called the "elimination method."
You can solve a system of linear equations by turning the coefficients of the variables into a square matrix, as noted in Hoffman and Kunze's "Linear Algebra." For example, 2x+4y=0 3x+2y=2 yields the matrix 2__4 3__2. Here, the underscores are merely spacers. If A is the matrix above, X is the variable vector (x y), and B is the vector (0 2), then the matrix multiplication representation is AX=B. Therefore, if A* is the inverse of A, then X equals A*B, the matrix product of A* and B. Excel can solve for the elements of X if they are unique.
A linear equation is usually written as aX + bY = c, where "a," "b" and "c" stand for known numeric coefficients and "X" and "Y" are variables. A solution of such equations is a set of variables that satisfies the expression. The addition method of solving linear equations aims to eliminate one of the variables by adding up those equations modified in a certain way. As an example, solve the following linear equations: 12X -- 7Y = 15 and -3X + 4Y = 27.
Linear equations describe the relationship between two or more variables. In a linear equation, all variables are multiplied by constants. There are no exponents other than the implicit first power. Linear equations can be used to determine the value of a variable if the corresponding values of the other variables are known. In coordinate geometry, if two separate points are known, a linear equation can be written expressing the line along which those points lie.
Linear equations are especially useful to describe the relationship between two or more variables. In a linear equation, each variable is modified only by a coefficient (a number by which it is multiplied). Variables are directly related if an increase in one corresponds to an increase in another. Conversely, variables are inversely related if an increase in one corresponds to a decrease in another. Linear equations will always graph onto the coordinate plane as straight lines.
A system of linear equations can be solved as long as you are presented with at least as many distinct equations as you have unknown variables. In such a case, there are two primary methods of solving the system: substitution and combination. Substitution consists of solving one equation in terms of one variable and substituting the solution for that variable into one of the other equations. Combination consists of adding or subtracting entire equations. In any given situation, either approach will yield the same results, usually with the same amount of work.
The slope-intercept form of a linear equation, in contrast to the standard form, is a powerful tool for graphing and visualizing linear relationships. In the slope-intercept form, the relevant aspects of a line are immediately apparent, and the relationship between the two variables is easy to visualize. Graphing, then, becomes a task of either mapping out the line's slope and y-intercept or plotting individual points using the slope-intercept equation.
Making equations that are linear and exponential in variables, or unknowns, can be demonstrated in the down-to-earth terms of how a bank account grows with interest. Linear equations can be made to represent simple interest. Exponential equations can be made to represent compound interest.
Linear equations contain multiple variables and have to be simultaneous in order to be solved. Simultaneous linear equations can written in the general form as a1X + b1Y = c1 and a2X + b2Y = c2. A solution of such a system is a set of variables "X" and "Y" that simultaneously satisfy these equations. The letters "a1," "a2," "b1," "b2," "c1" and c2 abbreviate the numeric coefficients in equations. One method of solving such linear equations is to use Cramer's rules. It involves calculations of determinants for three matrixes that are comprised from the equation coefficients. This method is…
A linear equation is a polynomial equation of the first degree. In other words, it is an algebraic equation in which each term is either a constant or the product of a constant and the first power of a single variable.
Operations research is an interdisciplinary branch of applied math. The central objective is optimization--to do the best under the given circumstances. The applications are nearly encyclopedic, including agricultural planning, distribution of goods, manpower allocation, industrial engineering, placement of extra shielding on warplanes, traffic control and much more.
Quantifying data or processes that are non-linear in nature can be a matter of curve-fitting or looking at the underlying process that created the data to fit an appropriate nonlinear model. A system is nonlinear if the relationship between its input and output cannot be described by a straight line.
Gauss-Jordan elimination is a version of Gaussian elimination in solving systems of linear equations. The variable coefficients, instead of merely being reduced to a triangle, are reduced to a diagonal. This eliminates the need for successive substitution, which simplifies coding for computer programming.
A linear equation is an algebraic equation in which each term is a constant or the product of a constant and a variable. Linear equations appear with great regularity because so many measurable quantities are proportional to other quantities as in related linearly. Furthermore, linear equations can be helpful first approximations of computationally prohibitive nonlinear phenomena.
Simultaneous equations are two or more equations with multiple variables. A solution of those equations is a set of variables that simultaneously satisfy all equations. The linear equations are generally given as "Y=aX+b," while non-linear equations can be any expressions not described as linear (e.g. "5X^3-7Y^2=21"). "X" and "Y denote equation variables and numbers before variables (e.g. "5" and "-7") are called coefficients. As an example, we will solve two non-linear simultaneous equations with two variables "X" and "Y". 2X^2+5Y^2=30 and 3X^2-4Y=20.
Linear equations are one of the basic fundamentals of upper-level mathematics, ranging from middle school algebra to college-level calculus. They have many mathematical applications, as the formulas that many are familiar with are just linear equations with specific variables.
Y-intercept is a characteristic of linear equation plots that are always straight lines. Linear equations are given in the form: y = ax + b. The letters a and b denote the equation coefficients. Y-intercept is the y-coordinate of the point where the plot crosses the y-axis. Another important characteristic is the plot slope that is numerically equal to the coefficient a. In the steps below, we consider calculation of the y-intercept in two common cases. In the first case, a linear equation is defined with a slope and a point with coordinates x1 and y1. In the second one,…
A slope and Y-intercept are characteristics of linear equation plots. Such plots are always straight lines while linear equations are given in the form: Y=aX+b; "a" and "b" are coefficients. Y-intercept is the Y-coordinate of the point where the plot crosses the Y-axis. The slope is the ratio between Y and X-coordinate differences for any two points that belong to the plot. Thus, slope=(Y2-Y1)/(X2-X1), and X1,Y1 and X2,Y2 are coordinates of the first and the second points, respectively. Two such points unambiguously define a linear equation. As an example, calculate the slope and the Y-intercept if the graph passes through…
Linear equations are described by the expression Y=aX+b and are always plotted as a straight line. The coefficients "a" and "b" are fixed for a particular equation while variables "X" and "Y" are any numbers that satisfy this equation. Any 2 points with coordinates (X1, Y1) and (X2, Y2), which are crossed by the linear equation graph (line), define this equation. Hence, the coefficients "a" and "b" can be expressed using coordinates of those points. As an example, write a linear equation if its graph passes through 2 points with coordinates X1 = 2, Y1 = 17 and X2 =…
