How To Calculate Algebra
Algebraic equations are simple to solve--or calculate for the variable (usually x). An equation represents a mathematical phrase of operations acting on x--those operations generally being addition, subtraction, multiplication and division. They can also involve squaring and cubing, for example. The phrase essentially tells you the order of operations. You can think of calculating for x as reversing the process to undo the operations to isolate x. If you have a quadratic equation, in which there is an x^2 and an x, you will need to either use the quadratic formula to calculate the value(s) of x or you would need to factor the equation.
For the purposes of this article, we will consider simple equations only, in which there is only one variable.
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Instructions
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Solving an Equation for X
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Eliminate any parenthesis by performing whatever operation is indicated. For example, say your original equation is 2(x - 6) = 3x + (4 - x)/2.
Eliminate parenthesis with the distributive property: 2x - 12 = 3x + 2 - x/2.
In this instance, you are simply carrying out an operation dictated by the math expression. -
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Combine like terms. This means, combine any constants or any variables that are already on the same side of the equation. Again, you are simply doing those additions or subtractions that are part of your problem's expressions. (You have 2 expressions with an equation--one on the left side and another on the right.)
In our example, there are no constants to combine (just yet) but we can combine the 3x and the -x/2 on the right side of the equation. To combine variable quantities, the variable must be the same (i.e. both are x) and then you simply add (or in this case subtract) the coefficient quantities, resulting in: 3 - 1/2 = 5/2 as the new coefficient for x. So we get: 2x - 12 = 5/2 x + 2.
Note: the x is not part of the denominator. You would write this on paper as the x being with the 5 in the numerator and have only the 2 in the denominator. -
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Move the constants to one side and the variable quantities to the other. Traditionally, x goes on the left and the constants to the right, although this is not necessary. Once you begin moving terms, you have stopped following the expression's operations and are now "undoing" them.
When a quantity moves to the opposite side of an equals sign (note that it is the equals, or equality of the two phrases, that makes this an equation) then the quantity must become opposite. In other words, a positive constant on one side becomes a negative constant on the other side while a negative constant moves across the equals to become positive.
What is actually happening is you are eliminating the constant on one side by performing an equal but opposite operation. So to move the -12 to the right, we are essentially adding 12 (the opposite of subtracting 12). As -12 and +12 cancel each other to zero, we no longer have the twelve on the left side. However, because this is an equation and because you have opted to do something that was not part of the expression (i.e. adding 12), you must treat both sides equally. So, if you add 12 to the left, you must also add 12 to the right.
2x -12 + 12 = 5/2 x + 2 + 12
This reduces to: 2x = 5/2 x + 14
But you could have simply thought of the entire process as being:
2x - 12 = 5/2 x + 2 goes to: 2x = 5/2 x + 2 + 12 which is: 2x = 5/2 x + 14 -
4
Move the variable quantities the same way. Since 5/2 x is a positive quantity on the right side, it becomes a negative quantity on the left. (Yes--you are actually subtracting 2/5 x from both sides of the equation)
Now you have: 2x - 5/2 x = 14
Again, you combine terms as we have two terms with x on the left side. In combining variable quantities we have the coefficients 2 - 5/2 resulting in -1/2. This gives us -x/2 = 14. -
5
Reduce the x coefficient to 1 once you have moved and combined your equation to a simple matter of a quantity of x equals a constant (which we now have.) Because a coefficient is a quantity multiplied to x, you cancel out the coefficient with division. Our coefficient is -1/2. To divide this by -1/2 is the same as multiplying by -2. (You could also have though of our x as having been divided by -2, in which case you would reverse the process by multiplying by -2. Either way you think of it, you arrive at the same solution.) But as was true in moving our values across the equals, what you do on one side to simplify the coefficient to 1 must be duplicated on the other. So if you multiply the left by -2, you must multiple the right by -2.
Thus, you have -2 * -x/2 = -2 * 14.
Since -2 * -1/2 = 1, giving us 1x (the quantity we seek) we now have x = -28.
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Tips & Warnings
In general, you perform all the simple operations that you can first. Use the distributive property (if you have parenthesis.) Perform any additions or subtractions you can to combine terms. Then begin reversing the process to undo the expressions. It is in this undoing process that you must always treat each side of the equation the same--i.e. add a quantity to both sides or multiply a quantity to both sides, for example. In undoing, you will, in general, undo with addition and subtraction first--until you can do no more. Then you begin undoing the expressions with multiplication or division. If you should have a simple quadratic, where your single variable is an x^2 term, the last operation you would perform on both sides of the equation would be to square root both sides--to undo the square operation.
