About Single-Variable Linear Equations
This article will tell you all the relevant information you need to know about linear equations in one variable.
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Definition
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A single-variable linear equation is an algebraic equation in which two numerical expressions with one variable of the first degree are set equal to one another. This type of equation can be modeled in the standard form Ax + B = C, where A, B and C are constant numerical values.
Solution
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There is only one solution to a single-variable linear equation. When you insert this numerical solution, sometimes called a root, in place of the variable it represents in the equation, it makes the equation true by making the left side of the equation equal the right side of the equation.
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Addition and Subtraction Properties
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You can use the addition and subtraction properties of equality to solve a single-variable linear equation by adding or subtracting the same value to or from both sides of the equation to isolate the variable. For instance, in the equation x - 4 = 6, you can add 4 to both sides of the equation to isolate x: x - 4 + 4 = 6 + 4. This leaves you with x on the left side of the equation and 10 on the right side; x = 10 is the solution to the equation.
The subtraction property can be applied the same way, as in the equation x + 4 = 6. Subtract 4 from both sides to isolate x and obtain the solution x = 2.
Multiplication and Division Properties
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You can use the multiplication and division properties of equality to solve a single-variable linear equation by multiplying or dividing both sides of the equation by the same number to isolate the variable and discover its value.
For instance, in the equation (2/5)x = 4, you can multiply both sides by 5/2, which cancels out the coefficient of the variable on the left side; any number multiplied by its reciprocal equals 1. On the right side you have (5/2)(4) = (20/2) = 10. Therefore, the root, or solution, is x = 10.
The division property can be applied the same way, as in the equation 4x = 16. Divide both sides by 4 to isolate x and obtain the solution of (16/4), which equals 4. Thus, x = 4.
Distributive Property and Combining Like Terms
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To solve single-variable linear equations such as 3 - 4(m - 4) = -2 + (1/2)m + 2.5m, we use two new procedures: the distributive property and combining like terms.
The order of algebraic operations states that equations are to be solved in a specific order: first parentheses, then exponents, then multiplication and division, then addition and subtraction, which is summarized by the acronym PEMDAS and the mnemonic phrase "Please excuse my dear Aunt Sally."
Thus, we first eliminate the parentheses on the left side of the equation by applying the distributive property and multiplying -4 by both m and -4. This yields the equation 3 - 4m + 16 = -2 + (1/2)m + 2.5 m."
Next we combine the like terms: the constants go with constants, and variables with variables. Moving the m's to one side of the equation and the constants to the other side of the equation by using the addition property of equality and then combining the like terms results in the equation 21 = 7m.
Use the division property of equality to divide both sides by 7 to isolate the variable m, and the solution is m = 3.
Real-Life Applications
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Single-variable linear equations can be used to model real-life applications, as in this question: "Patricio, a vacuum salesman, is paid a fixed rate of $500 per month, with the addition of a $2,000 commission per sale. How many vacuums does he have to sell per year to earn $104,000 per year?"
Begin by setting up the equation:
(12 months in a year) times A (fixed rate of pay per month, in $) + Bx (commission pay, where B is the commission per sale and x is the variable amount of sales) = C (dollars per year)
Simply plug in the values from the word problem, and solve:
12(500) + 2,000x = 104,000
6,000 + 2,000x = 104,000
2,000x = 98,000
x = 49
Solution: At a fixed rate of pay of $500 per month, Patricio will have to sell 49 vacuums per year to earn $104,000 annually.
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