Solving Systems of Three Equations with Elimination

Instructor: Mia Primas

Mia has taught math and science and has a Master's Degree in Secondary Teaching.

Solving systems of equations that have three equations and three variables can be a challenge. In this lesson, you will learn how to identify additive inverses that will help you to solve the system using the elimination method.

Identifying Additive Inverses

The systems of equations in this lesson contain three equations with three variables. The system is solved when we find a value for each variable that satisfies all three equations. There are many strategies that can be used to solve for the variables. In this lesson, we focus on the elimination method.

The goal of the elimination method is for one or more variables to get eliminated when the equations are added together. If we see that one equation has a term with the additive inverse of a term in another equation, those terms will be eliminated. Terms are considered additive inverses when their sum is zero. The following image shows examples of additive inverses. Notice that the terms contain the same variable and coefficient, but they have opposite signs.


Examples of additive inverses
sys1

Applying the Elimination Method


System of equations for Example 1
sys2

Notice that the x terms in Equation 1 and Equation 2 are additive inverses. When we add the equations together, the variable x will be eliminated, leaving the variables y and z. Unfortunately, we cannot solve an equation that has two variables. So we will have to continue to look for other patterns in the system that may be useful.


Adding equations 1 and 2
sys3

Equations 1 and 3 have additive inverses for the x terms and y terms so when we add them , both variables will be eliminated. The new equation, 3z = 3, has only one variable that we can solve for. Dividing by three on both sides, we are left with z = 1.


Adding equations 1 and 3
sys4

Now that we know the value of z, we can substitute it into our new equation of 2y - z = -11 and solve for y. Once we know the values of two variables, we can substitute those into one of the original equations and solve for the third variable.


Solution for example 1
sys5

The solution to the system is (2, -5, 1).


System of equations for example 2
sys6

The y terms in Equation 4 and Equation 5 are additive inverses, so let's start by adding those equations together.


Adding equations 4 and 5
sys7

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