Solving Systems with 3 Equations

By: Laney Emerson

What is a System of Equations?

A system of 3 equations is a situation in which you have to find the value of multiple variables given 3 equations. This can be useful when you are looking to find the intersection of 3 planes on a graph. Some ways to solve a system of equations include substitution, elimination, or by graphing. A system of equations can have 3 possible outcomes; one solution, no solution, or infinite solutions.

Solving a System of 3 Equations: One Solution

Step 1

Things to consider:
  • In the second equation y has a coefficient of 1 making it a good choice for elimination
  • Once y has been eliminated, there are only 2 variables and thus there is now a system of 2 equations
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Step 2

Things to consider:
  • After this step you will have the value of 2 variables
  • Elimination and substitution will be used
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Step 3

Things to consider:

  • This is the final step to solve these equations
  • The solution, called an ordered triple, is in the form of (x,y,z)
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Solving a System of 3 Equations: Infinite Solutions

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Solving a System of 3 Equations: No Solution

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Solving a system of 3 equations can be hard but it is best to go about the problem in steps! Be prepared for any of the possible outcomes and always show all your work! Because it is easy to make small sign or number mistakes, make sure you check your work.

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McGraw Hill Education- Algebra 2 Text Book

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