← Back to the Systems Of Equations worksheet
Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Standards: 8.EE.C.8a, 8.EE.C.8b, 8.EE.C.8c · Level: Grade 8
Problem 1 Is it a solution?
(a) 3 + 2 = 5 and 2(3) − 2 = 4: both true. (b) 4 + 1 = 5, but 2(4) − 1 = 7, not 4.
(a) Yes (b) No
Problem 2 Read the intersection
The lines cross where x = 2 and y = 3. Check: 2(2) − 1 = 3 and −2 + 5 = 3.
(2, 3)
Problem 3 Substitution, warm-up
Replace y by 2x: x + 2x = 12, so 3x = 12 and x = 4. Then y = 2(4) = 8. Check: 4 + 8 = 12.
(4, 8)
Problem 4 Elimination, warm-up
Add the equations: 2x = 12, so x = 6. Then 6 + y = 9 gives y = 3. Check: 6 − 3 = 3.
(6, 3)
Problem 5 Equal expressions
Both expressions equal y, so x + 6 = 4x − 3. Then 9 = 3x and x = 3. So y = 3 + 6 = 9. Check: 4(3) − 3 = 9.
(3, 9)
Problem 6 Estimate and verify
The lines cross at (3, 1). Check: 3 − 2 = 1 and −2(3) + 7 = 1.
(3, 1)
Problem 7 Substitute an expression
Replace x: (3y − 4) + 2y = 11, so 5y − 4 = 11 and 5y = 15. Then y = 3 and x = 3(3) − 4 = 5. Check: 5 + 6 = 11.
(5, 3)
Problem 8 Add to eliminate
Add: 4x = 16, so x = 4. From the second equation, 4 − 2y = 0, so y = 2. Check: 12 + 4 = 16.
(4, 2)
Problem 9 Multiply, then eliminate
Multiply the second equation by 3: 12x − 3y = 15. Add to the first: 14x = 28, so x = 2. Then 4(2) − y = 5, so y = 3. Check: 4 + 9 = 13.
(2, 3)
Problem 10 Scale one equation
Multiply the second equation by 2: 6x + 4y = 26. Subtract the first: 5x = 5, so x = 1. Then 1 + 4y = 21 gives y = 5. Check: 3 + 10 = 13.
(1, 5)
Problem 11 Parallel lines
Solve each for y: y = −23x + 2 and y = −23x + 4. The slopes match and the intercepts differ, so the lines are parallel. Also, 2x + 3y cannot equal both 6 and 12.
No solution
Problem 12 Same line
Rewrite the second equation: −2y = −10x + 8, so y = 5x − 4. It is the same line as the first equation, so every point on it is a solution.
Infinitely many solutions
Problem 13 Classify without solving
(a) Same slope 3, different intercepts: parallel. (b) Slopes −1 and 2 differ, so the lines cross once. (c) Doubling the first equation gives the second: same line.
(a) none (b) one (c) infinitely many
Problem 14 Two lines, one point
Set the expressions equal: −2x + 9 = 3x − 6. Then 15 = 5x and x = 3. So y = −2(3) + 9 = 3. Check: 3(3) − 6 = 3.
(3, 3)
Problem 15 Bake sale
Equations: x + y = 40 and 2x + 3.5y = 95. From the first, x = 40 − y. Substitute: 2(40 − y) + 3.5y = 95, so 80 + 1.5y = 95 and y = 10. Then x = 30. Check: 60 + 35 = 95.
30 muffins and 10 brownies
Problem 16 Trail meet-up
Ana: y = 3x + 2. Ben: y = 4x. Set 4x = 3x + 2, so x = 2 and y = 8. Check: 3(2) + 2 = 8.
After 2 hours, 8 miles from the trailhead
Problem 17 Concert tickets
Let x = adult tickets and y = child tickets. x + y = 150 and 12x + 7y = 1,330. Substitute x = 150 − y: 1,800 − 12y + 7y = 1,330, so −5y = −470 and y = 94. Then x = 56. Check: 672 + 658 = 1,330.
56 adult and 94 child tickets
Problem 18 Rectangle dimensions
Let y = length and x = width. Then 2y + 2x = 56, which gives y + x = 28, and y = 2x + 1. Substitute: 2x + 1 + x = 28, so 3x = 27 and x = 9. Then y = 19.
Width 9 ft; length 19 ft
Problem 19 Build a system
One choice: x + y = 3 and x − y = 5. Check (4, −1): 4 + (−1) = 3 and 4 − (−1) = 5. The slopes −1 and 1 differ, so the lines cross exactly once.
Example: x + y = 3 and x − y = 5
Problem 20 Car rental
A: y = 0.20x + 35. B: y = 0.35x + 20. Set them equal: 15 = 0.15x, so x = 100 and the cost is $55. At 250 miles: A costs 35 + 50 = $85 and B costs 20 + 87.50 = $107.50.
100 miles ($55); Company A is cheaper at 250 miles