How to Solve Equations (2 variables, substitution)

To solve algebraic equations in 2 variables, one easy method is the substitution method.

This way of solving systems of equations will help you in a variety of math problems, including word problems, and equations of lines in the xy-plane.

Difficulty: Moderately Easy

Instructions

1. Write down the two equations, in any form, and pick the one that looks easier to work with first. For example, let's say the two equations are:
   
   4x - y = -16
   3x = 18 - 5y
   
   We will pick the second equation to work with since one variable (the x term) is almost already isolated. Now, take that equation and completely isolate one of the variables (ie, solve for it). If we pick the x term we get:
   
   x = 6 - (5/3)y
   
   Don't worry that the other variable (the y term) is still around. It is supposed to stay.

2. Take the what you obtained in Step 1 (x = 6 - (5/3)y) and substitute it in the other equation (4x - y = -16). That is, in the equation 4x - y = -16, you replace the "x" with "6 - (5/3)y" so that you no longer have any x term. Let's see how it works:
   
   4(6 - (5/3)y) - y = -16
   24 - (20/3)y - y = -16
   
   Combine the y terms into (23/3)y:
   
   24 - (23/3)y = -16
   -(23/3)y = -40
   (23/3)y = 40
   
   Multiply both sides by the reciprocal of the y coefficient
   
   y = 120/23.

3. Take the value you obtained in Step 2 and plug it back into one of the other equations to solve for the other variable. It doesn't matter with equation you pick for the last step. We'll use 4x - y = -16
   
   4x - 120/23 = -16
   4x = -248/23
   x = -62/23.