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.

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.