Worksheets · Algebra 2

Systems of three equations worksheet for Algebra 2

Eliminate one variable from two different pairs, which leaves an ordinary two-equation system, then work back. Some systems have no solution or infinitely many, and some are ticket word problems.

Also taught in Precalculus, at its own level and with its own examples.

Select your difficulty

Pick more than one for a sheet that mixes them.

The preview can look cramped on a phone. The PDF and printed sheet come out normal.

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. Solve a system of three equations

    5x−4y+3z=383x−y−2z=13−3x−2y+2z=8
  2. x−2y+3z=−5−x+y+3z=4−2x+y+12z=9
  3. 2x+4y−2z=−405x+4y+5z=−38x−5y−5z=21
  4. 2x+3y−3z=−7x+3y−2z=−24x+9y−7z=−8
  5. 5x−y−5z=−11−4x+y−5z=−554x−y−3z=−1
  6. 2x−3y−3z=−32x+y−2z=−56x−y−7z=−12
  7. −x−4y−3z=−4−4x+4y−2z=−66−x+y−2z=−27
  8. 2x−3y−z=02x+3y+2z=18x+6y+5z=3
  9. 4x−4y+4z=0−x+2y+2z=−4x+y−2z=10
  10. 2x+y−2z=7−3x−2y+z=25x+3y−3z=5
  11. −3x+5y+5z=114x−5y−4z=−14−4x−y+4z=2
  12. x+2y−z=−4−2x−2y+z=4−5x−4y+2z=8
  13. −2x+y−z=−6x+2y+2z=15x+3y−z=9
  14. x−2y+z=7−3x+y+2z=24x−3y−z=5

  1. Solve word problems with three unknowns

    Sam invested $5,500: some in savings at 1%, some in bonds at 4% and the rest in stocks at 8% a year. The first year's interest was $308, and there was $900 more in stocks than in bonds. How much went into each?
  2. A basketball team scored 30 points on 2-point baskets, 3-point baskets and 1-point free throws, 18 scores in all. It made twice as many 2-point baskets as 3-point baskets. How many of each did it make?
  3. A jar holds 33 coins, all nickels, dimes and quarters, worth $3.05 in all. There are 9 more dimes than nickels. How many of each coin are in the jar?
  4. An aquarium sold 179 tickets on Saturday and took in $1014. Adult tickets are $8, student tickets $6 and child tickets $1. It sold twice as many student tickets as child tickets. How many of each were sold?
  5. A fund of $11,100 is split among savings (2%), bonds (6%) and stocks (9%) and earns $635 in a year. It has twice as much in bonds as in savings. How much is in each?
  6. A zoo charges $13 for adults, $5 for students and $2 for children. One day it sold equal numbers of student and child tickets and 208 tickets in all, for $1241. How many adult, student and child tickets were sold?

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. Solve a system of three equations

    5x−4y+3z=383x−y−2z=13−3x−2y+2z=8(2,−7,0)
  2. x−2y+3z=−5−x+y+3z=4−2x+y+12z=9No solution
  3. 2x+4y−2z=−405x+4y+5z=−38x−5y−5z=21(−4,−7,2)
  4. 2x+3y−3z=−7x+3y−2z=−24x+9y−7z=−8No solution
  5. 5x−y−5z=−11−4x+y−5z=−554x−y−3z=−1(4,−4,7)
  6. 2x−3y−3z=−32x+y−2z=−56x−y−7z=−12No solution
  7. −x−4y−3z=−4−4x+4y−2z=−66−x+y−2z=−27(7,−6,7)
  8. 2x−3y−z=02x+3y+2z=18x+6y+5z=3Infinitely many solutions
  9. 4x−4y+4z=0−x+2y+2z=−4x+y−2z=10(4,2,−2)
  10. 2x+y−2z=7−3x−2y+z=25x+3y−3z=5Infinitely many solutions
  11. −3x+5y+5z=114x−5y−4z=−14−4x−y+4z=2(−2,2,−1)
  12. x+2y−z=−4−2x−2y+z=4−5x−4y+2z=8Infinitely many solutions
  13. −2x+y−z=−6x+2y+2z=15x+3y−z=9(3,3,3)
  14. x−2y+z=7−3x+y+2z=24x−3y−z=5Infinitely many solutions

  1. Solve word problems with three unknowns

    Sam invested $5,500: some in savings at 1%, some in bonds at 4% and the rest in stocks at 8% a year. The first year's interest was $308, and there was $900 more in stocks than in bonds. How much went into each?$800 in savings, $1,900 in bonds, $2,800 in stocks
  2. A basketball team scored 30 points on 2-point baskets, 3-point baskets and 1-point free throws, 18 scores in all. It made twice as many 2-point baskets as 3-point baskets. How many of each did it make?6 two-point baskets, 3 three-point baskets, 9 free throws
  3. A jar holds 33 coins, all nickels, dimes and quarters, worth $3.05 in all. There are 9 more dimes than nickels. How many of each coin are in the jar?11 nickels, 20 dimes, 2 quarters
  4. An aquarium sold 179 tickets on Saturday and took in $1014. Adult tickets are $8, student tickets $6 and child tickets $1. It sold twice as many student tickets as child tickets. How many of each were sold?65 adult, 76 student, 38 child
  5. A fund of $11,100 is split among savings (2%), bonds (6%) and stocks (9%) and earns $635 in a year. It has twice as much in bonds as in savings. How much is in each?$2,800 in savings, $5,600 in bonds, $2,700 in stocks
  6. A zoo charges $13 for adults, $5 for students and $2 for children. One day it sold equal numbers of student and child tickets and 208 tickets in all, for $1241. How many adult, student and child tickets were sold?54 adult, 77 student, 77 child

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 and third to remove z: 2x + 3y = 8.
  2. Add the second and third to remove z: 3x + y = 5.
  3. Solve that pair: x = 1 and y = 2, then z = 3.

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.

What you can put on this worksheet

Three equations in three unknowns
−2x+y+2z=52x+y+2z=53x−3y+z=−1 → (0,1,2)
No solution or infinitely many
2x+3y−2z=1x+3y−3z=84x+9y−8z=13 → No solution
Word problems
Maya has $4.20 in nickels, dimes and quarters: 37 coins, with twice as many nickels as quarters. How many nickels, dimes and quarters does she have? → 20 nickels, 7 dimes, 10 quarters
Missing letters, solved forms, write one
Solve:
5x+4y−3z=32
−3x−5y−4z=2
2y=6 → (1,3,−5)

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.

You pick from 4: three equations in three unknowns, no solution or infinitely many, word problems and missing letters, solved forms, write one. Tick as many as you want and set how many of each, or let it spread them evenly.

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.