Algebra 1 Systems of Linear Equations Test Solutions and Scoring (PDF)

Algebra 1 math test solutions: Systems of Linear Equations – Maths PDF

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This systems test answers for Algebra 1 covers systems of linear equations (Common Core A-REI.C.5, A-REI.C.6, A-REI.D.11, A-CED.A.3), with a free PDF to print.

Use this key to grade the systems of equations test. Each solution shows the full work, and the scoring note explains how the points are awarded. The test totals 20 points. Give partial credit for correct steps, even when the final answer is wrong.

Algebra 1 · Common Core State Standards: A-REI.C.5, A-REI.C.6, A-REI.D.11, A-CED.A.3

Systems test answers: worked solutions

Question 1 – Check a solution [2 pts]

First: $3 + 2(-1) = 1$ true. Second: $3(3) – (-1) = 10$ true.

Yes

Scoring: 1 point for each equation checked correctly.

Question 2 – Solve by graphing [2 pts]

The lines meet at (2, 2).

The lines intersect at $(2, 2)$. Check: $2(2) – 2 = 2$ and $-2 + 4 = 2$.

$(2, 2)$

Scoring: 1 point for graphing both lines correctly; 1 point for the solution.

Question 3 – Substitution [2 pts]

Substitute: $3x + (2x – 3) = 12$, so $5x = 15$ and $x = 3$.

$y = 2(3) – 3 = 3$.

$(3, 3)$

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

Question 4 – Elimination [2 pts]

Add: $6x = 12$, so $x = 2$. Then $4(2) + 3y = 17$ gives $y = 3$.

$(2, 3)$

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

Question 5 – Elimination with multiplying [2 pts]

Multiply the first by 5 and the second by 2: $15x + 10y = 35$ and $4x – 10y = -16$. Add: $19x = 19$, so $x = 1$.

$3(1) + 2y = 7$ gives $y = 2$.

$(1, 2)$

Scoring: 1 point for a correct elimination step; 1 point for the solution.

Question 6 – Number of solutions [2 pts]

Lines for question 6b.

a) Rewrite the second: $2y = -4x + 6$, so $y = -2x + 3$. The same line: infinitely many solutions.

b) Equal slopes, different intercepts: parallel lines, no solution.

Scoring: 1 point for each part with reasoning.

Question 7 – Ticket sales [3 pts]

a) $s + a = 150$ and $5s + 8a = 930$.

b) Substitute $s = 150 – a$: $5(150 – a) + 8a = 930$, so $750 + 3a = 930$, $a = 60$, and $s = 90$.

c) $90 + 60 = 150$ and $5(90) + 8(60) = 450 + 480 = 930$.

90 student tickets and 60 adult tickets

Scoring: 1 point per part.

Question 8 – Break-even point [3 pts]

Break-even at 20 lawns.

a) $200 + 5x = 15x$.

b) $200 = 10x$, so $x = 20$. Revenue: $15(20) = 300$.

c) After 20 lawns, cost and revenue are both $300; the business breaks even.

20 lawns; $300

Scoring: 1 point per part.

Question 9 – Find the error [2 pts]

Check the second equation: $4 – 6 = -2$, not 2.

Add the equations: $2x = 12$, so $x = 6$. Then $y = 4$.

$(6, 4)$

Scoring: 1 point for showing the pair fails; 1 point for the correct solution.

Total score

Total

20 points

Key facts behind this systems test answers

Key facts

A solution of a system is an ordered pair that satisfies every equation.
Graphing: the solution is the intersection point of the lines.
Substitution: solve one equation for a variable and substitute into the other.
Elimination: add or subtract equations to cancel a variable.
Parallel lines give no solution; identical lines give infinitely many.

How to use this systems test answers

Score your test with the point values shown for each question. Then read the worked solution of every question where you lost points and note the method to review.

All Algebra 1 chapters: Algebra 1 math.

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