Systems of Equations Worksheet

23 printable problems: solve systems by graphing, substitution and elimination, spot systems with no solution or infinitely many, and set up word problems. Answer key with the key step for every problem.

23 problems · answer key included · free, no sign-up

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Teachers and parents: print the student copy for class or homework, or print with the answer key on a separate page. You can also link to this page from your course site or LMS; no account is needed. Problems get harder within each section.

A. Solve by graphing

Graph both lines on the same grid. The solution is the point where they cross.

1
${y = x + 1 y = - x + 5$
2
${y = 2 x - 3 y = - x + 3$
3
${y = \frac{1}{2} x + 1 y = - 2 x + 6$
4
${x + y = 4 x - y = 2$

B. Substitution

Solve one equation for a variable (if needed), substitute into the other, then back-substitute.

5
${y = 2 x x + y = 9$
6
${y = x - 4 3 x + y = 16$
7
${x = 3 y - 1 2 x + y = 12$
8
${y = - 2 x + 7 4 x - 3 y = 9$
9
${x + 2 y = 8 3 x - y = 3$
10
${y = \frac{1}{2} x + 2 x - 4 y = - 10$

C. Elimination

Add or subtract the equations (multiplying first if needed) so one variable cancels.

11
${x + y = 10 x - y = 4$
12
${2 x + 3 y = 12 2 x - y = 4$
13
${3 x + 2 y = 16 5 x - 2 y = 16$
14
${4 x + 3 y = 25 2 x + 5 y = 23$
15
${5 x - 2 y = 3 3 x + 4 y = 33$
16
${2 x - 3 y = - 1 3 x + 2 y = 18$

D. One, none or infinitely many?

Solve, or show that the system has no solution or infinitely many solutions.

17
${y = 2 x + 1 - 4 x + 2 y = 2$
18
${x + y = 5 2 x + 2 y = 7$
19
${y = 3 x - 2 y = 3 x + 4$

E. Word problems

Define two variables, write a system of two equations, and solve.

20
The sum of two numbers is 42 and their difference is 8. Find the numbers.
21
A school play sold 120 tickets for $800. Adult tickets cost $9 and student tickets cost $5. How many of each were sold?
22
3 pencils and 2 erasers cost $4.50. 5 pencils and 2 erasers cost $6.50. Find the price of each.
23
A jar has 30 dimes and quarters worth $5.10 in total. How many of each coin are there?

Answer key

Check your work after you finish each section.

A. Solve by graphing

1.
$(2, 3)$
Lines cross at $(2, 3)$
2.
$(2, 1)$
Lines cross at $(2, 1)$
3.
$(2, 2)$
Lines cross at $(2, 2)$
4.
$(3, 1)$
Rewrite as $y = - x + 4$ and $y = x - 2$ ; they cross at $(3, 1)$

B. Substitution

5.
$(3, 6)$
$x + 2 x = 9$ , so $x = 3$ , $y = 6$
6.
$(5, 1)$
$3 x + (x - 4) = 16$ , so $x = 5$ , $y = 1$
7.
$(5, 2)$
$2 (3 y - 1) + y = 12$ , so $y = 2$ , $x = 5$
8.
$(3, 1)$
$4 x - 3 (- 2 x + 7) = 9$ , so $x = 3$ , $y = 1$
9.
$(2, 3)$
$x = 8 - 2 y$ ; $3 (8 - 2 y) - y = 3$ , so $y = 3$ , $x = 2$
10.
$(2, 3)$
$x - 4 (\frac{1}{2} x + 2) = - 10$ , so $x = 2$ , $y = 3$

C. Elimination

11.
$(7, 3)$
Add: $2 x = 14$ , so $x = 7$ , $y = 3$
12.
$(3, 2)$
Subtract: $4 y = 8$ , so $y = 2$ , $x = 3$
13.
$(4, 2)$
Add: $8 x = 32$ , so $x = 4$ , $y = 2$
14.
$(4, 3)$
Multiply the second by 2 and subtract: $7 y = 21$ , so $y = 3$ , $x = 4$
15.
$(3, 6)$
Multiply the first by 2 and add: $13 x = 39$ , so $x = 3$ , $y = 6$
16.
$(4, 3)$
Multiply by 2 and 3, add: $13 x = 52$ , so $x = 4$ , $y = 3$

D. One, none or infinitely many?

17.
$Infinitely many solutions (same line)$
The second equation simplifies to $y = 2 x + 1$ , the same line
18.
$No solution (parallel lines)$
Doubling the first gives $2 x + 2 y = 10 \neq = 7$
19.
$No solution (parallel lines)$
Same slope, different intercepts

E. Word problems

20.
$25$ and $17$
System: $x + y = 42, x - y = 8$
21.
$50$ adult, $70$ student
System: $a + s = 120, 9 a + 5 s = 800$
22.
Pencil $1.00, eraser $0.75
System: $3 p + 2 e = 4.50, 5 p + 2 e = 6.50$
23.
$16$ dimes, $14$ quarters
System: $d + q = 30, 10 d + 25 q = 510$

How to solve a system of equations

Substitution. Solve one equation for one variable, for example $y = 2 x$ . Replace that variable in the other equation: $x + 2 x = 9$ , so $x = 3$ . Put $x = 3$ back into either equation to get $y = 6$ .

Elimination. Line the equations up in standard form. Multiply one or both so a variable has opposite coefficients, then add. In $x + y = 10$ and $x - y = 4$ , adding gives $2 x = 14$ , so $x = 7$ and $y = 3$ .

Graphing. Put both equations in slope-intercept form, graph them, and read the crossing point. Graphing is the best way to see why a system can have one solution, none (parallel lines) or infinitely many (the same line).

Always check by substituting your answer into both original equations.

Read the full Systems of Equations lesson with worked examples →

Frequently asked questions

What are the three ways to solve a system of equations?

Graphing (draw both lines and find where they cross), substitution (solve one equation for a variable and plug it into the other) and elimination (add or subtract the equations so one variable cancels). All three give the same answer; the best method depends on how the equations are written.

When should I use substitution instead of elimination?

Use substitution when one equation is already solved for a variable, like y = 2x + 1, or when a variable has a coefficient of 1. Use elimination when both equations are in standard form, Ax + By = C, especially when a variable already has matching or opposite coefficients.

How do you know if a system has no solution or infinitely many?

If solving leads to a false statement such as 0 = 3, the lines are parallel and there is no solution. If it leads to a true statement such as 0 = 0, both equations describe the same line and there are infinitely many solutions. In slope-intercept form: same slope with different intercepts means no solution; same slope and same intercept means infinitely many.

How can I check my answer?

Substitute the x and y values into both original equations. If both are true, the answer is correct. For example, (3, 6) checks in y = 2x because 6 = 2(3), and in x + y = 9 because 3 + 6 = 9.

Is there an answer key?

Yes. Every problem has its solution and the key step, and the graphing problems include a small graph of both lines. Print the student copy without answers, or print with the answer key on a separate page.

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