How to Determine Linear Equations

By Suzanne Akerman; Updated April 24, 2017
Put on your thinking cap and solve a simple form of an algebraic equation.

A linear equation is a simple algebraic equation including one or two variables, at least two expressions and an equals sign. These are the most basic equations in algebra, as they never require work with exponents or square roots. When a linear equation is graphed on a coordinate grid, it will always result in a straight line. A common form of a linear equation is y = mx + b; however, equations such as 4x = 12, .5 – n = 7 and 2300 = 300 + 28x are also linear equations.

How to Solve Linear Equations

Confirm that the equation you are attempting to solve is indeed a linear equation. If the problem includes an exponent or square root, it is not a linear equation. For example, 12 = 2x + 4 is linear. To solve a linear equation you must isolate the variable; this is also referred to as “solving for x.”

Combine like terms in the equation. For example, in the equation 3x + 7x = 30 you must first add 3x and 7x, since they are like terms. Similarly, for 68 = 12 – 4 + 5x, the 12 and the 4 must be combined. In the example 12 = 2x + 4, there are no like terms to combine.

Eliminate expressions from the equation by performing mathematic operations that retain the equality of both sides of the equation. For the example 12 = 2x + 4, subtract 4 from each side of the equation. Never perform an operation on only one side, or your equation will no longer be equal. Eliminating the 4 from both sides of the equation using the “addition of the opposite” principle results in the equation 8 = 2x.

Isolate the variable further. Do as many mathematic operations to both sides of the equation as it takes to get x by itself on one side of the equals sign. In the case of linear equations containing two variables, your result will be x in terms of y. For instance, x = 5y; these equations cannot be solved further without additional information. In the example 8 = 2x, both sides of the equation must be divided by 2 in order to eliminate the 2 on the right side of the equals sign. The result is 4 = x.

Place the variable on the left side of the equals sign. Rather than 4 = x, report your solution as x = 4. Check your work by using the answer you got for x in the original equation. In the example problem 12 = 2x + 4, this would be 12 = 2(4) + 4. This results in 12 = 12, so the answer is correct.

About the Author

Suzanne Akerman began writing in 2000. She has worked as a consultant at Pacific Lutheran University's Writing Center and her works have been published in the creative arts journal "Saxifrage." Akerman holds a Bachelor of Arts in English and a Master of Arts in education from Pacific Lutheran University.