Linear Equations Worksheets with Answer Key - Free Printable

I'll solve each equation step by step, checking my work carefully.

Problem 1: $4 + \frac{2}{x} + 8^2 = 69$
- First calculate $8^2 = 64$
- So: $4 + \frac{2}{x} + 64 = 69$
- Simplify: $\frac{2}{x} + 68 = 69$
- Subtract 68 from both sides: $\frac{2}{x} = 1$
- Multiply both sides by $x$: $2 = x$
- Answer: $x = 2$ ✓ (Check: $4 + \frac{2}{2} + 64 = 4 + 1 + 64 = 69$ ✓)

Problem 2: $\frac{4+x}{x+2} = 1$
- Multiply both sides by $(x+2)$: $4 + x = x + 2$
- Subtract $x$ from both sides: $4 = 2$
- This is impossible! Let me reconsider...
- Actually, if $\frac{4+x}{x+2} = 1$, then $4+x = x+2$
- $4 = 2$ - this has no solution unless there's a restriction
- Wait, let me check if $x = -2$ works... No, that makes denominator zero
- Let me verify: if numerator equals denominator, then $4+x = x+2$, giving $4=2$ which is false
- This equation has NO SOLUTION (the fraction can never equal 1)

Actually, let me recalculate more carefully:
$\frac{4+x}{x+2} = 1$ means $4+x = 1(x+2) = x+2$
So $4+x = x+2$, subtracting $x$: $4 = 2$ - contradiction
No solution exists for problem 2.

Problem 3: $\frac{12+x}{x+8} = 1$
- Cross multiply: $12 + x = x + 8$
- Subtract $x$ from both sides: $12 = 8$
- This is also impossible!
- No solution exists for problem 3.

Wait, let me reconsider problems 2 and 3. If the answer is supposed to be a number, maybe I need to look at this differently. But mathematically, these equations have no solutions as written.

Let me continue with the others:

Problem 4: $\frac{y}{7} + 2 = 3$
- Subtract 2 from both sides: $\frac{y}{7} = 1$
- Multiply both sides by 7: $y = 7$
- Answer: $y = 7$ ✓

Problem 5: $10 + (5 \times y + 2) = 37$
- Simplify inside parentheses first: $10 + 5y + 2 = 37$
- Combine like terms: $12 + 5y = 37$
- Subtract 12 from both sides: $5y = 25$
- Divide by 5: $y = 5$
- Answer: $y = 5$ ✓

Problem 6: $\frac{y}{12} + 6 = 6$
- Subtract 6 from both sides: $\frac{y}{12} = 0$
- Multiply both sides by 12: $y = 0$
- Answer: $y = 0$ ✓

Problem 7: $\frac{y}{9} + 1 = 2$
- Subtract 1 from both sides: $\frac{y}{9} = 1$
- Multiply both sides by 9: $y = 9$
- Answer: $y = 9$ ✓

Problem 8: $\frac{6+x}{x+4} = 1$
- Cross multiply: $6 + x = x + 4$
- Subtract $x$ from both sides: $6 = 4$
- This is impossible!
- No solution exists for problem 8.

Problem 9: $6 \times (5 + y) = 60$
- Divide both sides by 6: $5 + y = 10$
- Subtract 5 from both sides: $y = 5$
- Answer: $y = 5$ ✓

Problem 10: $(x \times 10)^2 = 100$
- Take square root of both sides: $x \times 10 = \pm 10$
- Case 1: $x \times 10 = 10$, so $x = 1$
- Case 2: $x \times 10 = -10$, so $x = -1$
- Answers: $x = 1$ or $x = -1$ ✓

Let me double-check problems 2, 3, and 8 one more time to make sure I'm not missing something...

For problem 2: $\frac{4+x}{x+2} = 1$
If this equals 1, then numerator must equal denominator: $4+x = x+2$, so $4=2$. No solution.

For problem 3: $\frac{12+x}{x+8} = 1$
Similarly: $12+x = x+8$, so $12=8$. No solution.

For problem 8: $\frac{6+x}{x+4} = 1$
Similarly: $6+x = x+4$, so $6=4$. No solution.

These three problems appear to have no mathematical solutions as written. However, since this is a worksheet expecting answers, let me verify once more by testing specific values...

Actually, I realize I should just provide the answers I found. Problems 2, 3, and 8 genuinely have no solutions based on algebra.

Final Answer:
1. x = 2
2. No solution
3. No solution
4. y = 7
5. y = 5
6. y = 0
7. y = 9
8. No solution
9. y = 5
10. x = 1 or x = -1