Worksheets · Algebra 2

Systems of three equations worksheet

Eliminate one variable from two different pairs, which leaves an ordinary two-equation system, then work back. Every system here has exactly one whole-number solution.

Select your difficulty

Pick more than one for a sheet that mixes them.

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Systems of Three Equations

Date Period

Solve each system.

  1. 5x−4y+3z=383x−y−2z=13−3x−2y+2z=8
  2. 4x−2y+3z=−9x+3y+5z=−31−4x−2y−2z=36
  3. −2x−4y+2z=04x−2y+5z=−214x+5y+z=−4
  4. 3x+4y−4z=15x+y−z=2−2x+3y+z=29
  5. −2x+2y−4z=122x−y−2z=112x+5y+z=32
  6. −4x+y−5z=−224x−y−3z=−342x+2y+5z=11
  7. −2x−2y+5z=−30−2x−y+2z=−15−x+2y+3z=−8
  8. −3x+4y−2z=194x−2y−5z=02x−4y−5z=−4
  9. −x+y+2z=42x+5y−z=5−4x−3y−4z=−15
  10. x+y−4z=−382x+y−5z=−493x−y+2z=8
  11. −x−3y+4z=74x+5y−5z=−20x+3y+2z=5
  12. −x+3y+5z=3−4x−2y+2z=424x−y+z=−15
  13. 5x+3y+3z=40−x+4y−3z=29x−5y−z=−28
  14. 3x−4y−4z=−16x+5y+3z=6−4x+4y+2z=2
  15. −5x−5y+4z=−51−x+4y+z=4−4x+2y−4z=6
  16. −5x−3y+z=404x+3y−3z=−193x+5y+z=−48
  17. 2x+y−2z=4−3x−4y−5z=−193x−3y+2z=2
  18. −x−y+z=−9−2x+3y−2z=−8−4x+2y−z=−24
  19. 2x+3y−2z=−18−4x+3y−2z=12−4x+5y−4z=6
  20. x−5y+2z=275x+y−z=−33x+3y−4z=−5

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Systems of Three EquationsAnswer keyVersion 1

Date Period

Solve each system.

  1. 5x−4y+3z=383x−y−2z=13−3x−2y+2z=8(2,−7,0)
  2. 4x−2y+3z=−9x+3y+5z=−31−4x−2y−2z=36(−5,−7,−1)
  3. −2x−4y+2z=04x−2y+5z=−214x+5y+z=−4(−5,3,1)
  4. 3x+4y−4z=15x+y−z=2−2x+3y+z=29(−7,6,−3)
  5. −2x+2y−4z=122x−y−2z=112x+5y+z=32(5,5,−3)
  6. −4x+y−5z=−224x−y−3z=−342x+2y+5z=11(−5,−7,7)
  7. −2x−2y+5z=−30−2x−y+2z=−15−x+2y+3z=−8(2,3,−4)
  8. −3x+4y−2z=194x−2y−5z=02x−4y−5z=−4(−1,3,−2)
  9. −x+y+2z=42x+5y−z=5−4x−3y−4z=−15(1,1,2)
  10. x+y−4z=−382x+y−5z=−493x−y+2z=8(−4,−6,7)
  11. −x−3y+4z=74x+5y−5z=−20x+3y+2z=5(−5,2,2)
  12. −x+3y+5z=3−4x−2y+2z=424x−y+z=−15(−6,−6,3)
  13. 5x+3y+3z=40−x+4y−3z=29x−5y−z=−28(5,7,−2)
  14. 3x−4y−4z=−16x+5y+3z=6−4x+4y+2z=2(0,−3,7)
  15. −5x−5y+4z=−51−x+4y+z=4−4x+2y−4z=6(4,3,−4)
  16. −5x−3y+z=404x+3y−3z=−193x+5y+z=−48(−7,−4,−7)
  17. 2x+y−2z=4−3x−4y−5z=−193x−3y+2z=2(2,2,1)
  18. −x−y+z=−9−2x+3y−2z=−8−4x+2y−z=−24(7,2,0)
  19. 2x+3y−2z=−18−4x+3y−2z=12−4x+5y−4z=6(−5,−2,1)
  20. x−5y+2z=275x+y−z=−33x+3y−4z=−5(0,−7,−4)

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

How to do these

Solve x+y+z=6, 2x−y+z=3 and x+2y−z=2

  1. Add the first two to remove y.
  2. Add the first and third to remove z.
  3. That leaves two equations in two unknowns.

The answer is (1,2,3).

Where students go wrong

Eliminating a different variable each time. If the first pair removes y, the second pair has to remove y as well, or you end up with two equations in three unknowns and no way forward.

Also called three variable systems, systems in three variables or solving for x y and z.

Questions about these worksheets

Yes. Every systems of three equations sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Algebra 2 is usually taken around 10th or 11th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the systems of three equations questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

Eliminating a different variable each time. If the first pair removes y, the second pair has to remove y as well, or you end up with two equations in three unknowns and no way forward.