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

Question 1: Check an ordered pair (2 points)

\(2(3)+(-1)=5\) is true, but \(3-(-1)=4\) is not equal to 5.

No, it is not a solution.

Scoring: 1 point for checking each equation correctly; the conclusion must match.

Question 2: Read a graph (3 points)

1. The lines cross at \((2,2)\).
2. \(\frac{1}{2}(2)+1=2\) and \(-2+4=2\). Both equations are true.

Scoring: 1 point for the point; 1 point for each equation checked.

Question 3: Solve by graphing (2 points)

The line \(y=x+1\) passes through \((0,1)\) and \((2,3)\). The line \(y=-2x+7\) passes through \((1,5)\) and \((2,3)\).

The solution is \((2,3)\).

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

Question 4: Substitution (2 points)

Substitute: \(2x+4x-5=7\), so \(6x=12\) and \(x=2\). Then \(y=4(2)-5=3\).

Solution: \((2,3)\).

Scoring: 1 point for \(x=2\); 1 point for \(y=3\).

Question 5: Elimination (2 points)

Subtract the second equation from the first: \(6y=18\), so \(y=3\). Then \(3x-6=0\) gives \(x=2\).

Solution: \((2,3)\). Check: \(3(2)+4(3)=18\).

Scoring: 1 point for eliminating a variable correctly; 1 point for the final pair.

Question 6: Multiply both equations (3 points)

Multiply the first by 2 and the second by 3: \(6x+4y=16\) and \(6x+15y=-6\).

Subtract: \(-11y=22\), so \(y=-2\).

Then \(3x+2(-2)=8\) gives \(3x=12\) and \(x=4\).

Solution: \((4,-2)\). Check: \(2(4)+5(-2)=-2\).

Scoring: 1 point for scaling the equations; 1 point for \(y=-2\); 1 point for \(x=4\).

Question 7: How many solutions? (2 points)

1. The first equation gives \(y=2x-4\). Same slope 2, different intercepts: no solution.
2. Dividing the first by 2 gives \(2x+y=3\), so \(y=-2x+3\). Same line: infinitely many solutions.

Scoring: 1 point per part for the correct count with a valid reason.

Question 8: School store (2 points)

Let \(n\) be notebooks and \(p\) pens: \(3n+4p=11.50\) and \(3n+2p=9.50\).

Subtract: \(2p=2\), so \(p=1\). Then \(3n+2=9.50\) gives \(n=2.50\).

A notebook costs $2.50 and a pen costs $1.00.

Scoring: 1 point for a correct system; 1 point for both prices.

Question 9: Gym memberships (2 points)

\(45+25m=75+20m\) gives \(5m=30\), so \(m=6\). Cost: \(45+25(6)=195\) and \(75+20(6)=195\).

The gyms cost the same after 6 months, $195.

Scoring: 1 point for the equation; 1 point for months and cost.

⬇ Download the Grade 8 Systems of Linear Equations test answer key (PDF, 105 KB)

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