Precalculus cramer's rule worksheet

Three determinants and two divisions. The bottom one is built from the coefficients; each top one replaces a column with the answers.

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Pick more than one for a sheet that mixes them.



Sigma Prep · sigmaprep.io/worksheets Name Cramer's Rule Date Period Solve each system using Cramer's rule. $\begin{aligned} 11 x + 4 y & = - 16 \\ 8 x - 11 y & = 197 \end{aligned}$ $\begin{aligned} - 5 x - 2 y & = 5 \\ - x - 9 y & = 1 \end{aligned}$ $\begin{aligned} - 5 x + 9 y & = - 55 \\ - 3 x + 7 y & = - 49 \end{aligned}$ $\begin{aligned} 5 x + 11 y & = - 70 \\ - 9 x - 5 y & = - 22 \end{aligned}$ $\begin{aligned} - 8 x + 2 y & = 8 \\ - x - 11 y & = 91 \end{aligned}$ $\begin{aligned} - 8 x + 11 y & = - 89 \\ 11 x - 2 y & = 83 \end{aligned}$ $\begin{aligned} 6 x + 8 y & = 18 \\ - 8 x + 3 y & = - 106 \end{aligned}$ $\begin{aligned} - 2 x - 4 y & = 10 \\ 6 x - 3 y & = 120 \end{aligned}$ $\begin{aligned} - 2 x + 7 y & = 96 \\ - 2 x - 2 y & = - 12 \end{aligned}$ $\begin{aligned} 2 x + 6 y & = 18 \\ - 9 x - 11 y & = - 1 \end{aligned}$ $\begin{aligned} - 3 x - 11 y & = - 163 \\ - 8 x + 2 y & = - 90 \end{aligned}$ $\begin{aligned} 9 x - 2 y & = - 94 \\ - 5 x + 5 y & = 95 \end{aligned}$ $\begin{aligned} 8 x - 5 y & = 49 \\ - 3 x - 11 y & = - 160 \end{aligned}$ $\begin{aligned} - x - 9 y & = 85 \\ - 3 x - 10 y & = 68 \end{aligned}$ $\begin{aligned} 5 x + 4 y & = 18 \\ - 6 x + 9 y & = 144 \end{aligned}$ $\begin{aligned} 8 x - 4 y & = - 76 \\ - 7 x + 5 y & = 53 \end{aligned}$ $\begin{aligned} 7 x + 2 y & = - 79 \\ - 9 x - 9 y & = 18 \end{aligned}$ $\begin{aligned} - 7 x + y & = 53 \\ - 2 x + 2 y & = 22 \end{aligned}$ $\begin{aligned} 3 x - 11 y & = 47 \\ x + 3 y & = - 11 \end{aligned}$ $\begin{aligned} 5 x + 2 y & = - 9 \\ - 11 x + 7 y & = 54 \end{aligned}$ 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.

$\begin{aligned} 11 x + 4 y & = - 16 \\ 8 x - 11 y & = 197 \end{aligned}$
$\begin{aligned} - 5 x - 2 y & = 5 \\ - x - 9 y & = 1 \end{aligned}$
$\begin{aligned} - 5 x + 9 y & = - 55 \\ - 3 x + 7 y & = - 49 \end{aligned}$
$\begin{aligned} 5 x + 11 y & = - 70 \\ - 9 x - 5 y & = - 22 \end{aligned}$
$\begin{aligned} - 8 x + 2 y & = 8 \\ - x - 11 y & = 91 \end{aligned}$
$\begin{aligned} - 8 x + 11 y & = - 89 \\ 11 x - 2 y & = 83 \end{aligned}$
$\begin{aligned} 6 x + 8 y & = 18 \\ - 8 x + 3 y & = - 106 \end{aligned}$
$\begin{aligned} - 2 x - 4 y & = 10 \\ 6 x - 3 y & = 120 \end{aligned}$
$\begin{aligned} - 2 x + 7 y & = 96 \\ - 2 x - 2 y & = - 12 \end{aligned}$
$\begin{aligned} 2 x + 6 y & = 18 \\ - 9 x - 11 y & = - 1 \end{aligned}$
$\begin{aligned} - 3 x - 11 y & = - 163 \\ - 8 x + 2 y & = - 90 \end{aligned}$
$\begin{aligned} 9 x - 2 y & = - 94 \\ - 5 x + 5 y & = 95 \end{aligned}$
$\begin{aligned} 8 x - 5 y & = 49 \\ - 3 x - 11 y & = - 160 \end{aligned}$
$\begin{aligned} - x - 9 y & = 85 \\ - 3 x - 10 y & = 68 \end{aligned}$
$\begin{aligned} 5 x + 4 y & = 18 \\ - 6 x + 9 y & = 144 \end{aligned}$
$\begin{aligned} 8 x - 4 y & = - 76 \\ - 7 x + 5 y & = 53 \end{aligned}$
$\begin{aligned} 7 x + 2 y & = - 79 \\ - 9 x - 9 y & = 18 \end{aligned}$
$\begin{aligned} - 7 x + y & = 53 \\ - 2 x + 2 y & = 22 \end{aligned}$
$\begin{aligned} 3 x - 11 y & = 47 \\ x + 3 y & = - 11 \end{aligned}$
$\begin{aligned} 5 x + 2 y & = - 9 \\ - 11 x + 7 y & = 54 \end{aligned}$



Sigma Prep · sigmaprep.io/worksheets Name Cramer's RuleAnswer keyVersion 1 Date Period Solve each system using Cramer's rule. $\begin{aligned} 11 x + 4 y & = - 16 \\ 8 x - 11 y & = 197 \end{aligned}$ $(4, - 15)$ $\begin{aligned} - 5 x - 2 y & = 5 \\ - x - 9 y & = 1 \end{aligned}$ $(- 1, 0)$ $\begin{aligned} - 5 x + 9 y & = - 55 \\ - 3 x + 7 y & = - 49 \end{aligned}$ $(- 7, - 10)$ $\begin{aligned} 5 x + 11 y & = - 70 \\ - 9 x - 5 y & = - 22 \end{aligned}$ $(8, - 10)$ $\begin{aligned} - 8 x + 2 y & = 8 \\ - x - 11 y & = 91 \end{aligned}$ $(- 3, - 8)$ $\begin{aligned} - 8 x + 11 y & = - 89 \\ 11 x - 2 y & = 83 \end{aligned}$ $(7, - 3)$ $\begin{aligned} 6 x + 8 y & = 18 \\ - 8 x + 3 y & = - 106 \end{aligned}$ $(11, - 6)$ $\begin{aligned} - 2 x - 4 y & = 10 \\ 6 x - 3 y & = 120 \end{aligned}$ $(15, - 10)$ $\begin{aligned} - 2 x + 7 y & = 96 \\ - 2 x - 2 y & = - 12 \end{aligned}$ $(- 6, 12)$ $\begin{aligned} 2 x + 6 y & = 18 \\ - 9 x - 11 y & = - 1 \end{aligned}$ $(- 6, 5)$ $\begin{aligned} - 3 x - 11 y & = - 163 \\ - 8 x + 2 y & = - 90 \end{aligned}$ $(14, 11)$ $\begin{aligned} 9 x - 2 y & = - 94 \\ - 5 x + 5 y & = 95 \end{aligned}$ $(- 8, 11)$ $\begin{aligned} 8 x - 5 y & = 49 \\ - 3 x - 11 y & = - 160 \end{aligned}$ $(13, 11)$ $\begin{aligned} - x - 9 y & = 85 \\ - 3 x - 10 y & = 68 \end{aligned}$ $(14, - 11)$ $\begin{aligned} 5 x + 4 y & = 18 \\ - 6 x + 9 y & = 144 \end{aligned}$ $(- 6, 12)$ $\begin{aligned} 8 x - 4 y & = - 76 \\ - 7 x + 5 y & = 53 \end{aligned}$ $(- 14, - 9)$ $\begin{aligned} 7 x + 2 y & = - 79 \\ - 9 x - 9 y & = 18 \end{aligned}$ $(- 15, 13)$ $\begin{aligned} - 7 x + y & = 53 \\ - 2 x + 2 y & = 22 \end{aligned}$ $(- 7, 4)$ $\begin{aligned} 3 x - 11 y & = 47 \\ x + 3 y & = - 11 \end{aligned}$ $(1, - 4)$ $\begin{aligned} 5 x + 2 y & = - 9 \\ - 11 x + 7 y & = 54 \end{aligned}$ $(- 3, 3)$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Cramer's RuleAnswer keyVersion 1

Date Period

Solve each system using Cramer's rule.

$\begin{aligned} 11 x + 4 y & = - 16 \\ 8 x - 11 y & = 197 \end{aligned}$ $(4, - 15)$
$\begin{aligned} - 5 x - 2 y & = 5 \\ - x - 9 y & = 1 \end{aligned}$ $(- 1, 0)$
$\begin{aligned} - 5 x + 9 y & = - 55 \\ - 3 x + 7 y & = - 49 \end{aligned}$ $(- 7, - 10)$
$\begin{aligned} 5 x + 11 y & = - 70 \\ - 9 x - 5 y & = - 22 \end{aligned}$ $(8, - 10)$
$\begin{aligned} - 8 x + 2 y & = 8 \\ - x - 11 y & = 91 \end{aligned}$ $(- 3, - 8)$
$\begin{aligned} - 8 x + 11 y & = - 89 \\ 11 x - 2 y & = 83 \end{aligned}$ $(7, - 3)$
$\begin{aligned} 6 x + 8 y & = 18 \\ - 8 x + 3 y & = - 106 \end{aligned}$ $(11, - 6)$
$\begin{aligned} - 2 x - 4 y & = 10 \\ 6 x - 3 y & = 120 \end{aligned}$ $(15, - 10)$
$\begin{aligned} - 2 x + 7 y & = 96 \\ - 2 x - 2 y & = - 12 \end{aligned}$ $(- 6, 12)$
$\begin{aligned} 2 x + 6 y & = 18 \\ - 9 x - 11 y & = - 1 \end{aligned}$ $(- 6, 5)$
$\begin{aligned} - 3 x - 11 y & = - 163 \\ - 8 x + 2 y & = - 90 \end{aligned}$ $(14, 11)$
$\begin{aligned} 9 x - 2 y & = - 94 \\ - 5 x + 5 y & = 95 \end{aligned}$ $(- 8, 11)$
$\begin{aligned} 8 x - 5 y & = 49 \\ - 3 x - 11 y & = - 160 \end{aligned}$ $(13, 11)$
$\begin{aligned} - x - 9 y & = 85 \\ - 3 x - 10 y & = 68 \end{aligned}$ $(14, - 11)$
$\begin{aligned} 5 x + 4 y & = 18 \\ - 6 x + 9 y & = 144 \end{aligned}$ $(- 6, 12)$
$\begin{aligned} 8 x - 4 y & = - 76 \\ - 7 x + 5 y & = 53 \end{aligned}$ $(- 14, - 9)$
$\begin{aligned} 7 x + 2 y & = - 79 \\ - 9 x - 9 y & = 18 \end{aligned}$ $(- 15, 13)$
$\begin{aligned} - 7 x + y & = 53 \\ - 2 x + 2 y & = 22 \end{aligned}$ $(- 7, 4)$
$\begin{aligned} 3 x - 11 y & = 47 \\ x + 3 y & = - 11 \end{aligned}$ $(1, - 4)$
$\begin{aligned} 5 x + 2 y & = - 9 \\ - 11 x + 7 y & = 54 \end{aligned}$ $(- 3, 3)$

How to do these

Solve $2 x + y = 7$ and $x - y = 2$

The coefficient determinant is 2(-1) - 1(1) = -3.
Replace the x column with 7 and 2 for the top of x.
Divide.

The answer is $(3, 1)$ .

Where students go wrong

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.

Also called solving systems with determinants or determinant method.

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.

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.

Precalculus cramer's rule worksheet

Select your difficulty

Cramer's Rule

Cramer's RuleAnswer keyVersion 1

How to do these

Where students go wrong

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