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## Solving a word problem with 3 unknowns using a linear equation

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## Solve Systems of Equations with Three Variables

By the end of this section, you will be able to:

- Determine whether an ordered triple is a solution of a system of three linear equations with three variables
- Solve a system of linear equations with three variables
- Solve applications using systems of linear equations with three variables

Before you get started, take this readiness quiz.

Solve a System of Linear Equations with Three Variables

The steps are summarized here.

Decide which variable you will eliminate.

Work with a pair of equations to eliminate the chosen variable.

Multiply one or both equations so that the coefficients of that variable are opposites.

- Add the equations resulting from Step 2 to eliminate one variable
- Repeat Step 2 using two other equations and eliminate the same variable as in Step 2.
- The two new equations form a system of two equations with two variables. Solve this system.
- Use the values of the two variables found in Step 4 to find the third variable.
- Write the solution as an ordered triple.
- Check that the ordered triple is a solution to all three original equations.

We check that the solution makes all three equations true.

Use equation (1) and (2) to eliminate z .

Use (4) and (5) to eliminate a variable.

We are left with a false statement and this tells us the system is inconsistent and has no solution.

Use equation (1) and (3) to eliminate x .

Use equation (1) and (2) to eliminate x again.

Solve Applications using Systems of Linear Equations with Three Variables

The fine arts department sold 75 adult tickets, 200 student tickets, and 75 child tickets.

The soccer team sold 200 adult tickets, 300 student tickets, and 100 child tickets.

## Key Concepts

In the following exercises, determine whether the ordered triple is a solution to the system.

In the following exercises, solve the system of equations.

In the following exercises, solve the given problem.

## Writing Exercises

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## Word problems relating 3 variable systems of equations

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## Topic Notes

- Solving 3 variable systems of equations by substitution
- Solving 3 variable systems of equations by elimination
- Solving 3 variable systems of equations with no or infinite solutions

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## IMAGES

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Systems of Equations Word Problems · mixture problem three variables · number system · How to solve a word problem with systems of equations.

This algebra video tutorial explains how to solve system of equations with 3 variables and with word problems. It contains two example word

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Solve a System of Linear Equations with Three Variables · Decide which variable you will eliminate. · Work with a pair of equations to eliminate the chosen

Check out StudyPug's tips & tricks on Word problems relating 3 variable systems of equations for Algebra.