# Three Different Ways to Solve a System of Linear Equations

By Misty Barton

A system of linear equations consists of multiple equations that use the same variable set, and which represent lines that exist on the same plane. A solution to a system of linear equations is the number or numbers that solve all of the equations at the same time, and represents the point at which the lines would cross if graphed. There are three ways to solve a system of equations that do not require the use of a calculator.

## Isolation and Substitution

When trying to solve a system of equations, begin by trying to isolate one of the variables. For example, if you are trying to solve the system y + 6x = 12 and 5x = 15 - 5y, the easiest variable to isolate is y in the first equation. To isolate y, move everything else in the problem to the opposite side of the equal sign. In the example, subtract 6x from both sides to get the rewritten equation y = 12 - 6x. Substitute this for y in the second equation. This will give you a new equation: 5x = 15 - 5(12 - 6x). Solve the new equation by isolating x; distribute the number outside the parentheses first. In the example, distributing the five would result in the equation 5x = 15 - 60 + 30x. Simplify the equation as much as possible. You can combine the 15 and -60 to get -45. Move the variables to one side of the equation and the numbers to the other side by performing opposite operations. In the example we would the resulting equation would be 45 = 25x. Solve for x by dividing by the coefficient in front of x. The example results in 1.8 = x.

To finish solving the equation, substitute the numerical solution for "x" back into one of the original equations and solve. In the example, 1.8 is substituted into y + 6x = 12 to get y + 6(1.8) = 12. This simplifies to y + 10.8 = 12. Getting y alone yields y = 1.2 So the solution for the system of equations is (1.8, 1.2).