Systems of equations worksheet

Every system here has exactly one solution and it is a whole-number point, because the point is chosen first and the equations are built around it. Tick substitution to get systems where one variable is already alone, or elimination for ones that line up.

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Pick more than one for a sheet that mixes them.

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Sigma Prep · sigmaprep.io/worksheets Name Systems of Equations Date Period Solve each system. Solve by substitution $\begin{aligned} x & = 5 y + 53 \\ - 5 x + 3 y & = - 45 \end{aligned}$ $\begin{aligned} y & = - 3 x + 14 \\ 6 x - 2 y & = 44 \end{aligned}$ $\begin{aligned} y & = - x - 11 \\ 5 x - 2 y & = - 27 \end{aligned}$ $\begin{aligned} x & = 5 y - 11 \\ - x + 3 y & = 3 \end{aligned}$ $\begin{aligned} y & = - 4 x - 16 \\ 4 x - 6 y & = 40 \end{aligned}$ $\begin{aligned} y & = - 4 x - 40 \\ x + 4 y & = - 40 \end{aligned}$ $\begin{aligned} x & = - 2 y + 6 \\ 6 x + 2 y & = - 14 \end{aligned}$ $\begin{aligned} y & = - 2 x + 2 \\ - 7 x + 2 y & = - 40 \end{aligned}$ $\begin{aligned} x & = 3 y + 32 \\ x - 6 y & = 62 \end{aligned}$ $\begin{aligned} x & = - 4 y - 36 \\ - x + 7 y & = - 74 \end{aligned}$ Solve by elimination $\begin{aligned} - x - 2 y & = - 8 \\ - 4 x + 4 y & = - 44 \end{aligned}$ $\begin{aligned} 2 x + 4 y & = 10 \\ 7 x + 8 y & = 29 \end{aligned}$ $\begin{aligned} 2 x + 6 y & = 0 \\ - 3 x + 18 y & = 81 \end{aligned}$ $\begin{aligned} 5 x - 5 y & = 40 \\ x + 15 y & = 8 \end{aligned}$ $\begin{aligned} - 3 x + 3 y & = - 36 \\ - 5 x - 9 y & = 10 \end{aligned}$ $\begin{aligned} - 7 x + 7 y & = 84 \\ - 2 x + 14 y & = 96 \end{aligned}$ $\begin{aligned} - 4 x - 7 y & = 81 \\ - 7 x - 21 y & = 203 \end{aligned}$ $\begin{aligned} 3 x - 4 y & = 11 \\ 6 x + 8 y & = - 10 \end{aligned}$ $\begin{aligned} - 5 x - 4 y & = 24 \\ x + 8 y & = 24 \end{aligned}$ $\begin{aligned} - 3 x + 4 y & = 62 \\ - 5 x - 8 y & = - 14 \end{aligned}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Systems of Equations

Date Period

Solve each system.

Solve by substitution
$\begin{aligned} x & = 5 y + 53 \\ - 5 x + 3 y & = - 45 \end{aligned}$
$\begin{aligned} y & = - 3 x + 14 \\ 6 x - 2 y & = 44 \end{aligned}$
$\begin{aligned} y & = - x - 11 \\ 5 x - 2 y & = - 27 \end{aligned}$
$\begin{aligned} x & = 5 y - 11 \\ - x + 3 y & = 3 \end{aligned}$
$\begin{aligned} y & = - 4 x - 16 \\ 4 x - 6 y & = 40 \end{aligned}$
$\begin{aligned} y & = - 4 x - 40 \\ x + 4 y & = - 40 \end{aligned}$
$\begin{aligned} x & = - 2 y + 6 \\ 6 x + 2 y & = - 14 \end{aligned}$
$\begin{aligned} y & = - 2 x + 2 \\ - 7 x + 2 y & = - 40 \end{aligned}$
$\begin{aligned} x & = 3 y + 32 \\ x - 6 y & = 62 \end{aligned}$
$\begin{aligned} x & = - 4 y - 36 \\ - x + 7 y & = - 74 \end{aligned}$

Solve by elimination
$\begin{aligned} - x - 2 y & = - 8 \\ - 4 x + 4 y & = - 44 \end{aligned}$
$\begin{aligned} 2 x + 4 y & = 10 \\ 7 x + 8 y & = 29 \end{aligned}$
$\begin{aligned} 2 x + 6 y & = 0 \\ - 3 x + 18 y & = 81 \end{aligned}$
$\begin{aligned} 5 x - 5 y & = 40 \\ x + 15 y & = 8 \end{aligned}$
$\begin{aligned} - 3 x + 3 y & = - 36 \\ - 5 x - 9 y & = 10 \end{aligned}$
$\begin{aligned} - 7 x + 7 y & = 84 \\ - 2 x + 14 y & = 96 \end{aligned}$
$\begin{aligned} - 4 x - 7 y & = 81 \\ - 7 x - 21 y & = 203 \end{aligned}$
$\begin{aligned} 3 x - 4 y & = 11 \\ 6 x + 8 y & = - 10 \end{aligned}$
$\begin{aligned} - 5 x - 4 y & = 24 \\ x + 8 y & = 24 \end{aligned}$
$\begin{aligned} - 3 x + 4 y & = 62 \\ - 5 x - 8 y & = - 14 \end{aligned}$



Sigma Prep · sigmaprep.io/worksheets Name Systems of EquationsAnswer keyVersion 1 Date Period Solve each system. Solve by substitution $\begin{aligned} x & = 5 y + 53 \\ - 5 x + 3 y & = - 45 \end{aligned}$ $(3, - 10)$ $\begin{aligned} y & = - 3 x + 14 \\ 6 x - 2 y & = 44 \end{aligned}$ $(6, - 4)$ $\begin{aligned} y & = - x - 11 \\ 5 x - 2 y & = - 27 \end{aligned}$ $(- 7, - 4)$ $\begin{aligned} x & = 5 y - 11 \\ - x + 3 y & = 3 \end{aligned}$ $(9, 4)$ $\begin{aligned} y & = - 4 x - 16 \\ 4 x - 6 y & = 40 \end{aligned}$ $(- 2, - 8)$ $\begin{aligned} y & = - 4 x - 40 \\ x + 4 y & = - 40 \end{aligned}$ $(- 8, - 8)$ $\begin{aligned} x & = - 2 y + 6 \\ 6 x + 2 y & = - 14 \end{aligned}$ $(- 4, 5)$ $\begin{aligned} y & = - 2 x + 2 \\ - 7 x + 2 y & = - 40 \end{aligned}$ $(4, - 6)$ $\begin{aligned} x & = 3 y + 32 \\ x - 6 y & = 62 \end{aligned}$ $(2, - 10)$ $\begin{aligned} x & = - 4 y - 36 \\ - x + 7 y & = - 74 \end{aligned}$ $(4, - 10)$ Solve by elimination $\begin{aligned} - x - 2 y & = - 8 \\ - 4 x + 4 y & = - 44 \end{aligned}$ $(10, - 1)$ $\begin{aligned} 2 x + 4 y & = 10 \\ 7 x + 8 y & = 29 \end{aligned}$ $(3, 1)$ $\begin{aligned} 2 x + 6 y & = 0 \\ - 3 x + 18 y & = 81 \end{aligned}$ $(- 9, 3)$ $\begin{aligned} 5 x - 5 y & = 40 \\ x + 15 y & = 8 \end{aligned}$ $(8, 0)$ $\begin{aligned} - 3 x + 3 y & = - 36 \\ - 5 x - 9 y & = 10 \end{aligned}$ $(7, - 5)$ $\begin{aligned} - 7 x + 7 y & = 84 \\ - 2 x + 14 y & = 96 \end{aligned}$ $(- 6, 6)$ $\begin{aligned} - 4 x - 7 y & = 81 \\ - 7 x - 21 y & = 203 \end{aligned}$ $(- 8, - 7)$ $\begin{aligned} 3 x - 4 y & = 11 \\ 6 x + 8 y & = - 10 \end{aligned}$ $(1, - 2)$ $\begin{aligned} - 5 x - 4 y & = 24 \\ x + 8 y & = 24 \end{aligned}$ $(- 8, 4)$ $\begin{aligned} - 3 x + 4 y & = 62 \\ - 5 x - 8 y & = - 14 \end{aligned}$ $(- 10, 8)$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Systems of EquationsAnswer keyVersion 1

Date Period

Solve each system.

Solve by substitution
$\begin{aligned} x & = 5 y + 53 \\ - 5 x + 3 y & = - 45 \end{aligned}$ $(3, - 10)$
$\begin{aligned} y & = - 3 x + 14 \\ 6 x - 2 y & = 44 \end{aligned}$ $(6, - 4)$
$\begin{aligned} y & = - x - 11 \\ 5 x - 2 y & = - 27 \end{aligned}$ $(- 7, - 4)$
$\begin{aligned} x & = 5 y - 11 \\ - x + 3 y & = 3 \end{aligned}$ $(9, 4)$
$\begin{aligned} y & = - 4 x - 16 \\ 4 x - 6 y & = 40 \end{aligned}$ $(- 2, - 8)$
$\begin{aligned} y & = - 4 x - 40 \\ x + 4 y & = - 40 \end{aligned}$ $(- 8, - 8)$
$\begin{aligned} x & = - 2 y + 6 \\ 6 x + 2 y & = - 14 \end{aligned}$ $(- 4, 5)$
$\begin{aligned} y & = - 2 x + 2 \\ - 7 x + 2 y & = - 40 \end{aligned}$ $(4, - 6)$
$\begin{aligned} x & = 3 y + 32 \\ x - 6 y & = 62 \end{aligned}$ $(2, - 10)$
$\begin{aligned} x & = - 4 y - 36 \\ - x + 7 y & = - 74 \end{aligned}$ $(4, - 10)$

Solve by elimination
$\begin{aligned} - x - 2 y & = - 8 \\ - 4 x + 4 y & = - 44 \end{aligned}$ $(10, - 1)$
$\begin{aligned} 2 x + 4 y & = 10 \\ 7 x + 8 y & = 29 \end{aligned}$ $(3, 1)$
$\begin{aligned} 2 x + 6 y & = 0 \\ - 3 x + 18 y & = 81 \end{aligned}$ $(- 9, 3)$
$\begin{aligned} 5 x - 5 y & = 40 \\ x + 15 y & = 8 \end{aligned}$ $(8, 0)$
$\begin{aligned} - 3 x + 3 y & = - 36 \\ - 5 x - 9 y & = 10 \end{aligned}$ $(7, - 5)$
$\begin{aligned} - 7 x + 7 y & = 84 \\ - 2 x + 14 y & = 96 \end{aligned}$ $(- 6, 6)$
$\begin{aligned} - 4 x - 7 y & = 81 \\ - 7 x - 21 y & = 203 \end{aligned}$ $(- 8, - 7)$
$\begin{aligned} 3 x - 4 y & = 11 \\ 6 x + 8 y & = - 10 \end{aligned}$ $(1, - 2)$
$\begin{aligned} - 5 x - 4 y & = 24 \\ x + 8 y & = 24 \end{aligned}$ $(- 8, 4)$
$\begin{aligned} - 3 x + 4 y & = 62 \\ - 5 x - 8 y & = - 14 \end{aligned}$ $(- 10, 8)$

How to do these

$\begin{aligned} x + 3 y & = 9 \\ 2 x - y & = 4 \end{aligned}$

The first equation gives x = 9 - 3y.
Put that into the second: 2(9 - 3y) - y = 4.
18 - 7y = 4, so y = 2, and then x = 3.

The answer is $(3, 2)$ .

Where students go wrong

Stopping after finding one variable. A system has two answers and they come as a pair, so finding y = 2 is halfway. The other one is not checking: putting the point back into both equations takes ten seconds and catches every arithmetic slip.

Also called simultaneous equations.

What you can put on this worksheet

Set up for substitution: $\begin{aligned} x & = - 2 y + 2 \\ 3 x + y & = 1 \end{aligned}$ → $(0, 1)$
Set up for elimination: $\begin{aligned} x - 3 y & = - 3 \\ - 4 x + 3 y & = 3 \end{aligned}$ → $(0, 1)$
Is the point a solution: Is $(0, 1)$ a solution of the system $x - 3 y = - 2$ and $- 4 x - 4 y = - 2$ ? → No, it fails both
One, none or infinitely many solutions: How many solutions: $y = 3 x - 5$ and $y = 3 x + 3$ → No solution
No solution or infinitely many, by substitution: Solve by substitution: ${\begin{cases} y = - 3 x - 6 \\ 6 x + 2 y = - 12 \end{cases}$ → Infinitely many solutions
No solution or infinitely many, by elimination: Solve by elimination: ${\begin{cases} 3 x - 2 y = 3 \\ 9 x - 6 y = 9 \end{cases}$ → Infinitely many solutions
Write a system with a given solution: Write a system of two equations whose solution is $(1, 1)$ → e.g.
$x + 3 y = 4$
$x + y = 2$

Questions about these worksheets

Yes. Every systems of 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 1 is usually taken around 8th or 9th 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 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 7: set up for substitution, set up for elimination, is the point a solution, one, none or infinitely many solutions, no solution or infinitely many, by substitution, no solution or infinitely many, by elimination and write a system with a given solution. Tick as many as you want and set how many of each, or let it spread them evenly.

Systems of equations worksheet

Select your difficulty

Solve by substitution

Solve by elimination

Solve by substitution

Solve by elimination

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these systems of equations worksheets free?+

Do the systems of equations worksheets come with an answer key?+

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What grade are these systems of equations worksheets for?+

Can I get a different version of the same systems of equations worksheet?+

Is this a free alternative to Kuta Software for systems of equations?+

Can I choose how hard the systems of equations questions are?+

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What do students get wrong with systems of equations?+