Systems of Equations Test Answer Key with Scoring – Grade 8

Question 1 Read and verify (2 points)

The lines cross at (4, 2). Check: 4 − 2 = 2 and −4 + 6 = 2.

(4, 2)

Scoring: 1 point for the pair, 1 point for both checks.

Question 2 Substitution (3 points)

Replace y: 2x + 4x − 5 = 13, so 6x = 18 and x = 3. Then y = 4(3) − 5 = 7. Check: 6 + 7 = 13.

(3, 7)

Scoring: 1 point for the substitution, 1 point for x, 1 point for y.

Question 3 Elimination (3 points)

Add the equations: 6x = 24, so x = 4. Then 16 + 3y = 25 gives y = 3. Check: 8 − 9 = −1.

(4, 3)

Scoring: 1 point for adding correctly, 1 point for x, 1 point for y.

Question 4 Classify the systems (3 points)

1. Same slope 5, different intercepts: no solution.
2. Multiply the second equation by 3: it becomes the first. Same line: infinitely many.
3. Slopes 4 and −1 differ: one solution.

(a) none (b) infinitely many (c) one

Scoring: 1 point per part.

Question 5 County fair tickets (3 points)

Let x = adult and y = child tickets. x + y = 120 and 9x + 5y = 860. Substitute x = 120 − y: 1,080 − 9y + 5y = 860, so −4y = −220 and y = 55. Then x = 65. Check: 585 + 275 = 860.

65 adult and 55 child tickets

Scoring: 1 point for the system, 1 point for y, 1 point for x.

Question 6 Pool memberships (3 points)

1. The lines cross at (3, 10).
2. After 3 visits both plans cost $10.
3. For 5 visits: A costs 2(5) + 4 = $14 and B costs 3(5) + 1 = $16, so Plan A is cheaper.

(a) (3, 10) (b) equal cost of $10 after 3 visits (c) Plan A

Scoring: 1 point per part.

Question 7 Is it a solution? (2 points)

2(−2) + 5 = 1: true. −2 − 5 = −7: true. Both are true.

Yes, (−2, 5) is a solution.

Scoring: 1 point per check.

Question 8 Parallel on purpose (1 point)

Keep the slope 2 and change the intercept, for example y = 2x + 5. Parallel lines never meet.

Example: y = 2x + 5

Scoring: 1 point for slope 2 with a different intercept.