Systems of Linear Equations Test Answer Key with Scoring – Algebra 1

Question 1 Is it a solution? (2 points)

First: 2(4) + (−3) = 5 ✓. Second: 4 − 3(−3) = 4 + 9 = 13 ✓.

Yes

Scoring: 1 point for each equation checked.

Question 2 Substitution (2 points)

4x + (3x − 2) = 12, so 7x = 14 and x = 2. Then y = 3(2) − 2 = 4.

Check: 8 + 4 = 12 ✓.

(2, 4)

Scoring: 1 point for the substituted equation, 1 point for the pair.

Question 3 Elimination (2 points)

Add: 8x = 24, so x = 3. Then 15 + 4y = 7 gives y = −2.

Check: 9 + 8 = 17 ✓.

(3, −2)

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

Question 4 Elimination with scaling (3 points)

Multiply the first by 4 and the second by 3: 8x + 12y = 76 and 15x − 12y = 39. Add: 23x = 115, so x = 5.

Then 10 + 3y = 19 gives y = 3.

Check: 25 − 12 = 13 ✓.

(5, 3)

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

Question 5 Number of solutions (2 points)

(a) The second equation gives y = −3x + 6, the same line: infinitely many solutions.

(b) Same slope 4, different intercepts: parallel lines, no solution.

Scoring: 1 point per part.

Question 6 Solution from a graph (3 points)

(a) The lines cross at (3, 1).

(b) y = x − 2: 3 − 2 = 1 ✓. y = −2x + 7: −6 + 7 = 1 ✓.

(c) The outputs are equal at input x = 3, and the common output is 1.

(3, 1)

Scoring: 1 point per part.

Question 7 Concession stand (3 points)

(a) 3h + 2s = 15 and 2h + 3s = 13.75.

(b) Multiply by 3 and 2: 9h + 6s = 45 and 4h + 6s = 27.5. Subtract: 5h = 17.5, so h = 3.50. Then 2s = 15 − 10.5 = 4.5, so s = 2.25.

(c) 3.50 + 2.25 = 5.75.

Hot dog $3.50, soda $2.25, together $5.75

Scoring: 1 point per part.

Question 8 Comparing plans (3 points)

(a) 15 + 4h = 3 + 7h.

(b) 12 = 3h, so h = 4. The cost is 31 dollars for both.

(c) Plan A: 15 + 24 = 39. Plan B: 3 + 42 = 45. Plan A is cheaper.

h = 4 hours; Plan A for 6 hours

Scoring: 1 point per part.