8th Grade Systems of Linear Equations Worksheet Solutions (PDF)
This systems of equations answers for 8th Grade covers systems of linear equations (Common Core 8.EE.C.8a, 8.EE.C.8b, 8.EE.C.8c), with a free PDF to print.
This answer key gives a full worked solution for each of the 21 problems on the systems of linear equations worksheet (Common Core 8.EE.C.8a, 8.EE.C.8b and 8.EE.C.8c). Compare each step with your own work, not only the final answer, and note any step where your method differed.
Grade 8 · Common Core State Standards: 8.EE.C.8a, 8.EE.C.8b, 8.EE.C.8c
Systems of equations answers: exercises 1 to 10
Exercise 1 – Check a solution
\(2 + 3 = 5\) is true. \(2 \cdot 2 – 3 = 1\) is true.
Yes
Exercise 2 – Not a solution
\(1 + 2 = 3\) is true, but \(1 – 2 = -1\), not 2.
No
Exercise 3 – Solve by graphing
Check: \(2 \cdot 2 – 1 = 3\) and \(-2 + 5 = 3\).
\((2, 3)\)
Exercise 4 – Graph two lines
Check: \(2 \cdot 2 – 2 = 2\) and \(-2 + 4 = 2\).
\((2, 2)\)
Exercise 5 – Substitution 1
\(x + 2x = 15\), so \(x = 5\).
\(y = 2 \cdot 5 = 10\).
\((5, 10)\)
Exercise 6 – Substitution 2
\(3x + (x + 3) = 19\), so \(4x = 16\) and \(x = 4\).
\(y = 4 + 3 = 7\).
\((4, 7)\)
Exercise 7 – Substitution 3
\(2(y – 2) + 3y = 21\), so \(5y – 4 = 21\) and \(y = 5\).
\(x = 5 – 2 = 3\).
\((3, 5)\)
Exercise 8 – Elimination 1
Add: \(2x = 22\), so \(x = 11\).
\(11 + y = 16\), so \(y = 5\).
\((11, 5)\)
Exercise 9 – Elimination 2
Add: \(5x = 20\), so \(x = 4\).
\(12 + y = 17\), so \(y = 5\).
\((4, 5)\)
Exercise 10 – Elimination with a multiplier
Multiply the second equation by 2: \(2x + 2y = 10\).
Subtract from the first: \(y = 2\). Then \(x = 3\).
\((3, 2)\)
Exercises 11 and up
Exercise 11 – Elimination, opposite terms
Add: \(8x = 24\), so \(x = 3\).
\(9 + 2y = 19\), so \(y = 5\).
\((3, 5)\)
Exercise 12 – Multiply both equations
Multiply the first by 3 and the second by 2: \(6x + 9y = 21\) and \(6x + 4y = 16\).
Subtract: \(5y = 5\), so \(y = 1\) and \(x = 2\).
\((2, 1)\)
Exercise 13 – Parallel lines
Both lines have slope \(-3\) but different intercepts, so they are parallel. Setting them equal gives \(2 = 7\), which is false.
No solution
Exercise 14 – Same line
The first equation is the second multiplied by 2, so they describe the same line.
Infinitely many solutions
Exercise 15 – Classify
Multiplying the first equation by 2 gives \(6x – 2y = 8\). The equations are equivalent.
Infinitely many solutions
Exercise 16 – School play tickets
Let \(a\) be adult tickets and \(s\) be student tickets: \(a + s = 50\) and \(12a + 8s = 480\).
Substitute \(s = 50 – a\): \(12a + 400 – 8a = 480\), so \(4a = 80\) and \(a = 20\).
20 adult tickets and 30 student tickets
Exercise 17 – Bake sale
\(m + c = 40\) and \(2m + 1.5c = 65\). Then \(c = 40 – m\).
\(2m + 60 – 1.5m = 65\), so \(0.5m = 5\) and \(m = 10\).
10 muffins and 30 cookies
Exercise 18 – Gym plans
\(10x + 25 = 15x + 15\), so \(10 = 5x\) and \(x = 2\).
Cost: \(10 \cdot 2 + 25 = 45\).
2 months, \$45
Exercise 19 – Boat and current
\(b + c = 18\) and \(b – c = 12\). Add: \(2b = 30\), so \(b = 15\).
\(c = 18 – 15 = 3\).
Boat 15 mph, current 3 mph
Exercise 20 – Coins
\(d + q = 30\) and \(10d + 25q = 450\) (in cents). Substitute \(d = 30 – q\): \(300 – 10q + 25q = 450\).
\(15q = 150\), so \(q = 10\) and \(d = 20\).
20 dimes and 10 quarters
Exercise 21 – Interpret the point
It is the only ordered pair that lies on both lines, so \(x = 4\) and \(y = 7\) make both equations true.
(4, 7) is the solution of the system
Key facts behind this systems of equations answers
- The solution of a system is the intersection point of the lines.
- Different slopes: one solution. Same slope, different intercepts: no solution. Same line: infinitely many.
- Substitution: solve one equation for a variable and substitute it into the other.
- Elimination: add or subtract the equations to cancel a variable.
- Always check the ordered pair in both equations.
How to use this systems of equations answers
Cover each solution, solve the exercise again, then uncover it and compare every line. If your answer differs, find the first step that changed and redo the exercise from there.
Download the systems of equations answers as a PDF with the button below and keep it with your notes.
All Grade 8 chapters: Grade 8 math.
