Linear Equations Test Answer Key with Scoring – Grade 8

Question 1 Variables on both sides (2 points)

Subtract 3x: 4x − 12 = 20. Add 12: 4x = 32. Divide by 4: x = 8.

x = 8

Scoring: 1 point for collecting variables, 1 point for x = 8.

Question 2 Parentheses (3 points)

Distribute: 8x − 12 = 3x + 18. Subtract 3x: 5x − 12 = 18. Add 12: 5x = 30. So x = 6.

x = 6

Scoring: 1 point for distributing correctly, 1 point for collecting terms, 1 point for x = 6.

Question 3 Fractions (3 points)

Subtract 15x: 15x + 3 = 7. Subtract 3: 15x = 4. Multiply by 5: x = 20.

x = 20

Scoring: 1 point for each of the three steps.

Question 4 Match the picture (3 points)

1. Distribute: 6x + 9 = 6x + 9, always true: panel B, infinitely many solutions.
2. Subtract 8x: −5 = 1, false: panel A, no solution.
3. 12 = 3x, so x = 4: panel C, one solution.

(a) B, infinitely many (b) A, none (c) C, x = 4

Scoring: 1 point per part.

Question 5 Frame perimeter (3 points)

2[(2x + 1) + (x + 3)] = 50, so 2(3x + 4) = 50 and 3x + 4 = 25. Then 3x = 21 and x = 7. Length = 15 and width = 10.

x = 7; length 15 in; width 10 in

Scoring: 1 point per part.

Question 6 Streaming plans (3 points)

30 + 12m = 6 + 15m. Subtract 12m: 30 = 6 + 3m. Subtract 6: 24 = 3m, so m = 8. Total: 30 + 12 · 8 = 126 and 6 + 15 · 8 = 126.

8 months; $126

Scoring: 1 point for the equation, 1 point for m = 8, 1 point for $126.

Question 7 Find Dev’s error (2 points)

Dev did not multiply the 4 by 3. Correct: 3x + 12 = 2x + 20. Subtract 2x: x + 12 = 20. So x = 8.

Error: 3 · 4 = 12, not 4. Correct answer: x = 8.

Scoring: 1 point for the error, 1 point for x = 8.

Question 8 Choosing k (1 point)

If k = 5, the x terms cancel and leave 2 = 9, which is false.

k = 5

Scoring: 1 point.