Worksheets · Precalculus

Systems of three equations worksheet for Precalculus

In Precalculus three-variable systems are solved with matrices: write the augmented matrix and row reduce, or use the inverse matrix when it exists. A row of all zeros means infinitely many solutions. A row of zeros with a nonzero number in the last column means no solution.

Also taught in Algebra 2, at its own level and with its own examples.

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Pick more than one for a sheet that mixes them.

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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

    8x+4y−7z=59−7x−4y+5z=−44−6x−8y−8z=42
  2. x−y−3z=−3−2x+3y−2z=−8−5x+8y−9z=−27
  3. 7x−8y−6z=647x+5y−7z=−595x−8y+4z=116
  4. 2x−3y−2z=23x+3y−z=2−x−6y−z=−2
  5. −6x+6y−5z=82x+5y+7z=−45−5x−7y−2z=15
  6. 2x−2y+z=−6−3x+y+z=−45x−3y=−2
  7. 5x+5y−7z=−277x−2y+7z=547x−5y+2z=21
  8. 3x+y−z=−1−3x+2y−3z=−7−6x+7y−10z=−21
  9. −4x+y+7z=41x−5y+8z=1305x+8y+z=−37
  10. x−y+3z=−4−2x+y+2z=−6−3x+y+7z=−11
  11. −x+6y+z=504x+7y−3z=45−2x+y+2z=12
  12. 3x+2y−3z=−52x−3y−z=−57x−4y−5z=−17
  13. x−4y+8z=286x+6y+8z=−22−4x−4y−3z=17
  14. 2x+2y−3z=4x+y+2z=−35x+5y+3z=−5

  1. Solve word problems with three unknowns

    A school play sold adult tickets at $30, student tickets at $12 and child tickets at $1. It sold 595 tickets in all, 3 more student tickets than twice the number of child tickets, and took in $8241. How many of each were sold?
  2. A basketball team scored 120 points on 2-point baskets, 3-point baskets and 1-point free throws, 68 scores in all. It made as many free throws as 2-point and 3-point baskets together. How many of each did it make?
  3. Sam invested $38,000: some in savings at 1%, some in bonds at 6% and the rest in stocks at 8% a year. The first year's interest was $2,060, and there was as much in stocks as in savings and bonds together. How much went into each?
  4. A parking meter was emptied of 132 coins (nickels, dimes and quarters) worth $14.25. It held 12 more nickels than twice the number of quarters. How many of each coin were there?
  5. A team's 56 scores (2-point baskets, 3-point baskets and 1-point free throws) gave it 111 points, with 7 more 2-point baskets than twice the number of 3-point baskets. Find how many of each the team made.
  6. A museum took in $3568 from 296 visitors. Adults pay $15, students $11 and children $6, and there were as many adults as students and children together. 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 2

Date Period

Solve each system.

  1. Solve a system of three equations

    8x+4y−7z=59−7x−4y+5z=−44−6x−8y−8z=42(5,−4,−5)
  2. x−y−3z=−3−2x+3y−2z=−8−5x+8y−9z=−27Infinitely many solutions
  3. 7x−8y−6z=647x+5y−7z=−595x−8y+4z=116(4,−9,6)
  4. 2x−3y−2z=23x+3y−z=2−x−6y−z=−2No solution
  5. −6x+6y−5z=82x+5y+7z=−45−5x−7y−2z=15(−4,3,−8)
  6. 2x−2y+z=−6−3x+y+z=−45x−3y=−2Infinitely many solutions
  7. 5x+5y−7z=−277x−2y+7z=547x−5y+2z=21(2,1,6)
  8. 3x+y−z=−1−3x+2y−3z=−7−6x+7y−10z=−21No solution
  9. −4x+y+7z=41x−5y+8z=1305x+8y+z=−37(5,−9,10)
  10. x−y+3z=−4−2x+y+2z=−6−3x+y+7z=−11No solution
  11. −x+6y+z=504x+7y−3z=45−2x+y+2z=12(−5,8,−3)
  12. 3x+2y−3z=−52x−3y−z=−57x−4y−5z=−17No solution
  13. x−4y+8z=286x+6y+8z=−22−4x−4y−3z=17(0,−5,1)
  14. 2x+2y−3z=4x+y+2z=−35x+5y+3z=−5Infinitely many solutions

  1. Solve word problems with three unknowns

    A school play sold adult tickets at $30, student tickets at $12 and child tickets at $1. It sold 595 tickets in all, 3 more student tickets than twice the number of child tickets, and took in $8241. How many of each were sold?151 adult, 297 student, 147 child
  2. A basketball team scored 120 points on 2-point baskets, 3-point baskets and 1-point free throws, 68 scores in all. It made as many free throws as 2-point and 3-point baskets together. How many of each did it make?16 two-point baskets, 18 three-point baskets, 34 free throws
  3. Sam invested $38,000: some in savings at 1%, some in bonds at 6% and the rest in stocks at 8% a year. The first year's interest was $2,060, and there was as much in stocks as in savings and bonds together. How much went into each?$12,000 in savings, $7,000 in bonds, $19,000 in stocks
  4. A parking meter was emptied of 132 coins (nickels, dimes and quarters) worth $14.25. It held 12 more nickels than twice the number of quarters. How many of each coin were there?78 nickels, 21 dimes, 33 quarters
  5. A team's 56 scores (2-point baskets, 3-point baskets and 1-point free throws) gave it 111 points, with 7 more 2-point baskets than twice the number of 3-point baskets. Find how many of each the team made.31 two-point baskets, 12 three-point baskets, 13 free throws
  6. A museum took in $3568 from 296 visitors. Adults pay $15, students $11 and children $6, and there were as many adults as students and children together. How many adult, student and child tickets were sold?148 adult, 92 student, 56 child

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

How to do these

x+y+z=6, 2x−y+z=3, x+2y−z=2

  1. Augmented matrix rows: [1 1 1 | 6], [2 -1 1 | 3], [1 2 -1 | 2].
  2. Row reduce to reduced row echelon form.
  3. Read x, y and z off the last column.

The answer is (1,2,3).

Where students go wrong

Stopping at row echelon form and misreading it. Either back-substitute from the bottom row up, or keep reducing until each row holds one variable.

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.

Precalculus is usually taken around 11th or 12th 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.