What is a solution of system of equations with 3 variables?

Solution for system of lines

Just as the solution system of lines is where those lines meet, a solution for a system of 3 variable equations (planes), is again, just where these planes meet.

Types of solutions for systems of planes (3 variable equations)

Example of how to solve a system of three variable equations using elimination.

Practice Problems

Problem 1)
Use elimination to solve the following system of three variable equations.

4x + 2y – 2z = 10

2x + 8y + 4z = 32

30x + 12y – 4z = 24

Solution

Problem 2)
Use elimination to solve the following system of three variable equations.

x - y + z = -1

x + y + z = 3

4x + 2y + z = 8

Problem 3)
Use elimination to solve the following system of three variable equations.

A)2x + 2y + 2z = -4

B) x + y + z = 3

C)4x + 2y + z = 8

Although you can indeed solve 3 variable systems using elimination and substitution as shown on this page, you may have noticed that this method is quite tedious. The most efficient method is to use matrices or, of course, you can use this online system of equations solver .