Solving Linear Equations: Grade 8 Math Test Answer Key – PDF Download

Question 1: Multi-step equations (2 points)

1. \(5x=25\), so \(x=5\).
2. \(3x-12=-21\), so \(3x=-9\) and \(x=-3\).

Scoring: 1 point per part. Accept a correct answer with minor arithmetic work not shown.

Question 2: Variables on both sides (3 points)

1. \(4x+5=25\), so \(4x=20\) and \(x=5\).
2. \(4x-8=x+10\), so \(3x=18\) and \(x=6\).
3. \(6x+3-4x=19\), so \(2x+3=19\), \(2x=16\) and \(x=8\).

Scoring: 1 point per part.

Question 3: Fractions and decimals (3 points)

1. \(\frac{2}{5}x=8\), so \(x=8\times \frac{5}{2}=\)20.
2. \(0.5x-0.5=2\), so \(0.5x=2.5\) and \(x=5\).
3. Multiply by 6: \(2(x+6)=3x-6\), so \(2x+12=3x-6\) and \(x=18\). Check: \(\frac{1}{3}(24)=8\) and \(\frac{1}{2}(18)-1=8\).

Scoring: 1 point per part.

Question 4: Number of solutions (3 points)

1. \(7x-2=7x-2\) is always true: infinitely many solutions.
2. \(3x+12=3x+9\) gives \(12=9\), false: no solution.
3. \(6x-4=2x+4\), so \(4x=8\): one solution, \(x=2\).

Scoring: 1 point per part: 1 for the correct classification with supporting work, solving for part (c).

Question 5: Choose the constant (2 points)

1. \(3(x+2)=3x+6\), so the sides match only when \(k=6\).
2. Any value other than 6 works, for example \(k=10\): \(3x+6=3x+10\) gives \(6=10\), which is false.

Scoring: 1 point per part. Accept any k other than 6 in part (b).

Question 6: Equal perimeters (2 points)

Rectangle: \(2(2x+1+5)=4x+12\). Square: \(4(x+7)=4x+28\).

Equation: \(4x+12=4x+28\). Subtract \(4x\): \(12=28\), which is false.

No solution. The square’s perimeter is always 16 more, so the perimeters are never equal.

Scoring: 1 point for a correct equation; 1 point for no solution with a reason.

Question 7: Savings plans (3 points)

1. Priya: \(64+8w\). Sam: \(28+14w\).
2. \(64+8w=28+14w\), so \(36=6w\) and \(w=6\). After 6 weeks.
3. \(64+8(6)=112\). Each has $112.

Scoring: 1 point per part.

Question 8: Find the error (2 points)

Noah did not multiply \(-2\) by 4 when he distributed. Correct: \(4(x-2)=4x-8\).

Then \(4x-8=2x+6\), so \(2x=14\) and \(x=7\). Check: \(4(5)=20\) and \(2(7)+6=20\).

Scoring: 1 point for identifying the distribution error; 1 point for the correct solution x = 7.

⬇ Download the Grade 8 Solving Linear Equations test answer key (PDF, 108 KB)

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