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Math Practice

Systems With Three Variables Practice Problems

Eliminate the same variable from two different pairs of equations, solve the 2 × 2 system that is left, then back-substitute. Triangular systems at the start let you practise back-substitution on its own.

  1. 1.

    Is a solution of , and ? Answer "yes" or "no".

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    ✓, ✓, ✓. Yes.

  2. 2.

    Solve the triangular system , , .

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    Bottom up: , then , then .

  3. 3.

    Solve , and .

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    Substitute: , so , , , .

  4. 4.

    Solve the triangular system , , .

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    , , .

  5. 5.

    Solve , , .

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    Add (1) and (3): . Add (2) and (3): . Then , , and from (1) .

  6. 6.

    Solve , , .

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    Add the last two: , so . Add the first two: , so . Then .

  7. 7.

    Solve , , .

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    (1) + (2): . 2·(2) + (3): , so . Subtract: , , , and .

  8. 8.

    Solve , , .

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    From the last two, and . Then gives , so , , .

  9. 9.

    Solve , , .

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    With : . Then gives , so , , .

  10. 10.

    Solve , , .

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    (1) + (2): . 2·(1) + (3): , so . Then , , and (2) gives .

  11. 11.

    Solve , , .

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    Subtracting the first equation from the second gives , which is false. No solution.

  12. 12.

    Three numbers add up to 30. The middle number is 2 more than the smallest, and the largest is twice the smallest. What is the largest number?

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    gives , so , the middle is 9 and the largest is 14.
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MPhil in Mathematics, Founder of Mathovia
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Systems With Three Variables Practice Problems

These problems follow the systems of equations with three variables lesson. Label your equations (1), (2), (3) so you can keep track of which pairs you combined.

The steps

  1. Eliminate one variable from a pair of equations.
  2. Eliminate the same variable from a different pair.
  3. Solve the resulting two-variable system.
  4. Back-substitute for the third variable and check all three equations.

Common mistakes to watch for

  • Eliminating different variables. Use two pairs of equations to eliminate the same variable, leaving a 2×2 system.
  • Reusing one equation twice. Each elimination must bring in the equation not yet used.
  • Arithmetic slips when scaling. Multiply every term of an equation, including the constant.
  • Not back-substituting fully. Once two variables are known, substitute both into an original equation to find the third.

Frequently Asked Questions

What does the answer to a 3-variable system look like?+

An ordered triple (x, y, z) that satisfies all three equations.

Which variable should I eliminate first?+

The one that cancels most easily, for example a variable with opposite coefficients in two equations. Eliminate the same variable from both pairs.

How do I type my answer?+

Type an ordered triple such as (1, 2, 3) or x = 1, y = 2, z = 3, or type no solution.

Calculators for this topic

Gaussian Elimination Calculator

More practice problems

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