Worksheets · Precalculus

Cramer's rule worksheet for Precalculus

Precalculus applies Cramer's rule to two-equation systems, which takes three 2x2 determinants, and to three-equation systems, which takes four 3x3 determinants. Each variable is its own determinant, with the constants swapped into that variable's column, divided by the coefficient determinant D.

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

Cramer's Rule

Date Period

Solve each system using Cramer's rule.

  1. Solve with Cramer's rule

    −6x+7y=−77−4x+3y=−43
  2. 2x−11y=144x+2y=−33
  3. 6x+3y=−27−8x+8y=0
  4. 10x−6y=14−10x+7y=−13
  5. −9x+5y=−120−10x−11y=−34
  6. −8x+7y=335x+2y=−78
  7. −5x+8y=124−x+9y=121
  8. −11x−2y=1459x−9y=−108
  9. 9x+2y=−131−2x−7y=75
  10. x+5y=−434x+4y=−44

  1. Solve three equations with Cramer's rule

    −4x+5y−4z=−26−4x−3y−4z=−105x+y+4z=18
  2. 5x+5y−3z=−2−2x−y+2z=2−4x−2y+5z=3
  3. −2x−y+4z=−22−4x−3y+5z=−365x+4y−4z=37
  4. −2x−3y+5z=−163x+2y−3z=5−4x−2y+5z=−6
  5. 5x+2y+5z=1−4x−y−2z=4−4x−2y+5z=27
  6. 4x+y+4z=−2−4x+y−4z=−24x−4y+5z=5
  7. −4x+5y+5z=−23−4x−y=15−4x+y+5z=−11
  8. x−y−2z=62x+4y−2z=10−3x−4y+z=−5
  9. x+y=−23x−y+5z=−245x+y−4z=−10
  10. −3x+y+3z=1−x+2y+5z=−9x+3y+3z=−3

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

Sigma Prep · sigmaprep.io/worksheets

Name

Cramer's RuleAnswer keyVersion 2

Date Period

Solve each system using Cramer's rule.

  1. Solve with Cramer's rule

    −6x+7y=−77−4x+3y=−43(7,−5)
  2. 2x−11y=144x+2y=−33(−5,−14)
  3. 6x+3y=−27−8x+8y=0(−3,−3)
  4. 10x−6y=14−10x+7y=−13(2,1)
  5. −9x+5y=−120−10x−11y=−34(10,−6)
  6. −8x+7y=335x+2y=−78(−12,−9)
  7. −5x+8y=124−x+9y=121(−4,13)
  8. −11x−2y=1459x−9y=−108(−13,−1)
  9. 9x+2y=−131−2x−7y=75(−13,−7)
  10. x+5y=−434x+4y=−44(−3,−8)

  1. Solve three equations with Cramer's rule

    −4x+5y−4z=−26−4x−3y−4z=−105x+y+4z=18(4,−2,0)
  2. 5x+5y−3z=−2−2x−y+2z=2−4x−2y+5z=3(−3,2,−1)
  3. −2x−y+4z=−22−4x−3y+5z=−365x+4y−4z=37(1,4,−4)
  4. −2x−3y+5z=−163x+2y−3z=5−4x−2y+5z=−6(−3,4,−2)
  5. 5x+2y+5z=1−4x−y−2z=4−4x−2y+5z=27(−2,−2,3)
  6. 4x+y+4z=−2−4x+y−4z=−24x−4y+5z=5(3,−2,−3)
  7. −4x+5y+5z=−23−4x−y=15−4x+y+5z=−11(−3,−3,−4)
  8. x−y−2z=62x+4y−2z=10−3x−4y+z=−5(−1,1,−4)
  9. x+y=−23x−y+5z=−245x+y−4z=−10(−4,2,−2)
  10. −3x+y+3z=1−x+2y+5z=−9x+3y+3z=−3(−3,4,−4)

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. The coefficient determinant is D = 7.
  2. Replace the x column with 6, 3, 2 to get Dx = 7; in the same way Dy = 14 and Dz = 21.
  3. Divide each by D: x = 1, y = 2, z = 3.

The answer is (1,2,3).

Where students go wrong

Putting the constants in the wrong column. For Dy the constants replace only the y coefficients, so in the example Dy has columns 1, 2, 1, then 6, 3, 2, then 1, 1, -1, and works out to 14.

Also called solving systems with determinants or determinant method.

What you can put on this worksheet

Two equations, solved with determinants
x−3y=−3−4x+3y=3 → (0,1)
Three equations
5x+5y−3z=−7
5x+5y−4z=−6
5x+4y+4z=−11 → (1,−3,−1)
Fractions, no or infinite solutions, messy forms
x−4y=−23
−7x−6y=−43 → (1,6)

Questions about these worksheets

Yes. Every cramer's rule 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 cramer's rule 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 3: two equations, solved with determinants, three equations and fractions, no or infinite solutions, messy forms. Tick as many as you want and set how many of each, or let it spread them evenly.

Replacing the wrong column, or forgetting to divide by the coefficient determinant at the end. Each variable gets its own top determinant, and both are divided by the same bottom one.