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

Grade 8 Linear Worksheet Answer Key: problems 1–5

Problem 1: Add then multiply

Subtract 7 from both sides: \(4x=24\). Divide both sides by 4: \(x=6\).

Check: \(4(6)+7=31\). Solution: \(x=6\).

Problem 2: Negative coefficient

Subtract 8: \(-3x=-18\). Divide by \(-3\): \(x=6\).

Check: \(-3(6)+8=-10\). Solution: \(x=6\).

Problem 3: Fraction coefficient

Add 3: \(\frac{x}{4}=8\). Multiply both sides by 4: \(x=32\).

Check: \(\frac{32}{4}-3=5\). Solution: \(x=32\).

Problem 4: Distribute first

Distribute: \(6x-3=45\). Add 3: \(6x=48\). Divide by 6: \(x=8\).

Check: \(3(2\cdot 8-1)=3(15)=45\). Solution: \(x=8\).

Problem 5: Variables on both sides

Subtract \(3x\) from both sides: \(2x+4=16\). Subtract 4: \(2x=12\). Divide by 2: \(x=6\).

Check: \(5(6)+4=34\) and \(3(6)+16=34\). Solution: \(x=6\).

Grade 8 Linear Worksheet Answer Key: problems 6–10

Problem 6: Collect the x terms

Subtract \(4x\): \(3x-2=13\). Add 2: \(3x=15\). Divide by 3: \(x=5\).

Check: \(7(5)-2=33\) and \(4(5)+13=33\). Solution: \(x=5\).

Problem 7: Variable with a negative

Add \(2x\) to both sides: \(9=3x+21\). Subtract 21: \(-12=3x\). Divide by 3: \(x=-4\).

Check: \(9-2(-4)=17\) and \(-4+21=17\). Solution: \(x=-4\).

Problem 8: Distribute and collect

Distribute: \(4x-12=2x+6\). Subtract \(2x\): \(2x-12=6\). Add 12: \(2x=18\). Divide by 2: \(x=9\).

Check: \(4(9-3)=24\) and \(2(9)+6=24\). Solution: \(x=9\).

Problem 9: Combine like terms

Distribute: \(3x+2x-8=17\). Combine like terms: \(5x-8=17\). Add 8: \(5x=25\). Divide by 5: \(x=5\).

Check: \(3(5)+2(1)=17\). Solution: \(x=5\).

Problem 10: Fraction equation

Subtract 1: \(\frac{2}{3}x=8\). Multiply by \(\frac{3}{2}\): \(x=8\times \frac{3}{2}=12\).

Check: \(\frac{2}{3}(12)+1=9\). Solution: \(x=12\).

Grade 8 Linear Worksheet Answer Key: problems 11–15

Problem 11: Fractions on both sides

Subtract \(\frac{1}{4}x\): \(\frac{1}{4}x-3=2\). Add 3: \(\frac{1}{4}x=5\). Multiply by 4: \(x=20\).

Check: \(\frac{1}{2}(20)-3=7\) and \(\frac{1}{4}(20)+2=7\). Solution: \(x=20\).

Problem 12: Decimal equation

Subtract \(0.1x\): \(0.3x+1.5=4.8\). Subtract 1.5: \(0.3x=3.3\). Divide by 0.3: \(x=11\).

Check: \(0.4(11)+1.5=5.9\) and \(0.1(11)+4.8=5.9\). Solution: \(x=11\).

Problem 13: Fraction times parentheses

Multiply both sides by \(\frac{4}{3}\): \(x-8=12\). Add 8: \(x=20\).

Check: \(\frac{3}{4}(20-8)=\frac{3}{4}(12)=9\). Solution: \(x=20\).

Problem 14: No solution

Subtract \(4x\) from both sides: \(6=9\). This statement is false for every value of \(x\).

No solution.

Problem 15: Infinitely many

Distribute: \(3x+6=3x+6\). Subtract \(3x\): \(6=6\). This statement is true for every value of \(x\).

Infinitely many solutions. Any number works.

Grade 8 Linear Worksheet Answer Key: problems 16–20

Problem 16: Sort the equations

1. \(5x-5=2x+10\), so \(3x=15\) and \(x=5\). One solution, \(x=5\).
2. Subtract \(7x\): \(-3=2\), which is false. No solution.
3. \(8x+6=8x+6\), which is always true. Infinitely many solutions.

Problem 17: Gym memberships

Jamal: \(15+4v\). Sofia: \(9+6v\). Set them equal: \(15+4v=9+6v\).

Subtract \(4v\): \(15=9+2v\). Subtract 9: \(6=2v\). So \(v=3\).

Cost: \(15+4(3)=27\). After 3 visits, both cost $27.

Problem 18: Taxi rates

Cab A: \(3.5+2m\). Cab B: \(2+2.5m\). Set equal: \(3.5+2m=2+2.5m\).

Subtract \(2m\): \(3.5=2+0.5m\). Subtract 2: \(1.5=0.5m\). Divide by 0.5: \(m=3\).

Fare: \(3.5+2(3)=9.5\). At 3 miles, both cost $9.50.

Problem 19: Rectangle perimeter

Perimeter \(=2(\text{length}+\text{width})\): \(2(2x+3+x)=48\). Combine: \(2(3x+3)=48\), so \(6x+6=48\).

Subtract 6: \(6x=42\). Divide by 6: \(x=7\).

Width \(=7\) ft, length \(=2(7)+3=17\) ft. Check: \(2(17+7)=48\). Answer: \(x=7\), 17 ft by 7 ft.

Problem 20: Saving up

Aisha: \(40+12w\). Diego: \(100+7w\). Set equal: \(40+12w=100+7w\).

Subtract \(7w\): \(40+5w=100\). Subtract 40: \(5w=60\). Divide by 5: \(w=12\).

Amount: \(40+12(12)=184\). After 12 weeks, each has $184.

Grade 8 Linear Worksheet Answer Key: problems 21–22

Problem 21: Number puzzle

Let \(n\) be the number: \(3(n+4)=5n-6\). Distribute: \(3n+12=5n-6\).

Subtract \(3n\): \(12=2n-6\). Add 6: \(18=2n\). Divide by 2: \(n=9\).

Check: \(3(13)=39\) and \(5(9)-6=39\). The number is 9.

Problem 22: Burning candles

Candle A: \(12-0.5h\). Candle B: \(9-0.25h\). Set equal: \(12-0.5h=9-0.25h\).

Add \(0.5h\): \(12=9+0.25h\). Subtract 9: \(3=0.25h\). Divide by 0.25: \(h=12\).

Height: \(12-0.5(12)=6\). After 12 hours, both are 6 inches tall.

How to use this grade 8 linear worksheet answer key

Read this grade 8 linear worksheet answer key after trying each problem yourself, and redo any problem you missed.

See every chapter on the Grade 8 math page.