Question

In your own words explain the steps to solve a system of linear equations with three variables by elimination.

Short answer

To solve a system of linear equations with three variables by elimination we first need to eliminate the same variable from the two equations. Then using two other equations we again eliminate the same variable. Now we get two new equations that form a system of two equations with two variables. We solve the system to get the values of the two variables and use these values to get the value of the third variable.

Step-by-step solution

Step 1. Definition

A linear equation with three variables, where $a,b,c$, and $d$are real numbers and $a,b$, and $c$are not all zero, is of the form $ax+by+cz=d$. Every solution to the equation is an ordered triple, $(x,y,z)$ that makes the equation true.

Step 2. Elimination Method

To solve a system of linear equations involving three variables, we generally use the same method we used with systems having two variables. The steps involved are

• We start with selecting two pairs of equations and in each pair, we eliminate the same variable.
• On doing that we will get a system of equations with only two variables.
• Then we solve the system of two equations with two variables.
• Now, we use the values of both variables we just found to go back to the original equation and find out the value of the third variable.
• We represent our answer as an ordered triple and then check our results.