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How to solve systems of 3 variable equations

Using Elimination

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

question
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)
types of solutions


Why are we going to use 3 planes?

question
If you want to solve an equation with 2 variables, you need 2 equations.
You can's solve x + y =1 , right? That's because you 1 equation with 2 variables
Similarly, if you have an equation with 3 variables, (3 planes), you're going to need 3 equations to solve it


Two important terms

Consistent question
Means that there is at least 1 instersection (solutions)
inonsistent question
Means that there are no intersections(solutions)


Ok, so how do we find that point of intersection?

Before attempting to solve systems of three variable equations using elimination, you should probably be comfortable solving 2 variable systems of linear equations using elimination .

Video Tutorial on using Elmination

Example 1
Example of how to solve a system of three variable equations using elimination. Steps to solve system of 3 equations
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
steps to solve 3 variable system by elimination

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 .