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Worksheets›Algebra 2›Systems of equations›Systems of three linear equations

Systems of Three Linear Equations Worksheets

Free printable worksheets with answer keys. Pick a level, preview the questions, then open the sheet to swap or delete any single question.

Algebra 2CCSS HSA.REI.C.6Answer key includedPDF · Print
Open in generatorStart with an easy sheet

Easy: Small whole-number solutions

  1. 1.x + 2y − z = 42x − y + z = 13x + y + z = 12
  2. 2.2x + y + z = 17x − y + z = 5x + 2y − z = 7
  3. 3.x + 2y − z = 42x − y + z = 9x + y + z = 7
  4. 4.x + 2y − z = 122x − y + z = 7x + y + z = 10
  5. 5.2x + y + z = 9x − y + z = −2x + 2y − z = 9
  6. 6.2x + y + z = 12x − y + z = 2x + 2y − z = 6
Preview shows 6 of 12 questions. Every sheet is new; open it to change any question.

Worksheet variations

Use two pairs of equations to eliminate the same variable, which leaves two equations in two variables. Solve that smaller system, then substitute back into any original equation to find the third variable. Check the answer in all three equations.

Easy2x + y + z = 14x − y + z = 2x + 2y − z = 7Small whole-number solutionsMediumx + y − z = 4x − 2y + z = 13x + y + z = −6Negative solutionsHard3x − y + 2z = 13x + y + z = 42x − y − z = 11Larger coefficients

How to solve it: a worked example

Solve the system. Write the answer as (x, y, z).

  1. Startx + 2y − z = 92x − y + z = 4x + y + z = 9
  2. Eliminate x from the second and third equations−5y + 3z = −14−y + 2z = 0
  3. Solve the two-variable systemy = 4, z = 2
  4. Substitute back to find xx = 3
  5. Answer(3, 4, 2)

Common mistakes to watch for

  • Eliminating a different variable from each pair.
  • Substituting into only one equation and not checking the others.
  • Sign errors when subtracting whole equations.

Tip: tick “Step-by-step answer key” in the generator to print each step for marking.

Related worksheets

Operations with complex numbersPowers of iQuadratic equations with complex solutionsDividing polynomials (long and synthetic division)The remainder and factor theorems

FAQ

What is the strategy for three equations?

Use two different pairs of equations to eliminate the same variable, leaving a system of two equations in two variables. Solve that, then substitute back to find the third variable.

How do you check the answer?

Substitute (x, y, z) into all three original equations. Checking only one or two can miss an arithmetic slip.

Is every worksheet different?

Yes. Numbers are generated each time, and the answer key is calculated for that sheet. Swap any single question and its answer updates too.

Can I mix systems of three linear equations with other skills?

Yes. In the generator, add more skills; each gets its own section on the sheet with its own instructions.

Is it free to print?

Yes. Generating and printing from the browser is free with no sign-up. PDF downloads need a free account.