Solve linear equations to find the unknown variable in this Math Monks worksheet.

Let’s solve each problem step by step. I’ll go one at a time, show the work clearly, and check my answer before writing it down.

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Problem 1:
$-\frac{9}{5}x = -45$

Step 1: To isolate x, multiply both sides by the reciprocal of $-\frac{9}{5}$, which is $-\frac{5}{9}$.

$x = -45 \times -\frac{5}{9}$

Step 2: Multiply:

$-45 \times -\frac{5}{9} = \frac{225}{9} = 25$

Check: Plug x = 25 back in:

$-\frac{9}{5}(25) = -45 → -45 = -45 ✓$

✔ Answer: 25

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Problem 2:
$\frac{x}{3} - \frac{x}{5} = 2$

Step 1: Find common denominator for 3 and 5 → 15

Rewrite: $\frac{5x}{15} - \frac{3x}{15} = 2$

Step 2: Combine: $\frac{2x}{15} = 2$

Step 3: Multiply both sides by 15: $2x = 30$

Step 4: Divide by 2: $x = 15$

Check: $\frac{15}{3} - \frac{15}{5} = 5 - 3 = 2 ✓$

✔ Answer: 15

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Problem 3:
$\frac{4x + 5}{6} = \frac{7}{2}$

Step 1: Cross-multiply: $2(4x + 5) = 6 \cdot 7$

→ $8x + 10 = 42$

Step 2: Subtract 10: $8x = 32$

Step 3: Divide by 8: $x = 4$

Check: $\frac{4(4)+5}{6} = \frac{21}{6} = \frac{7}{2} ✓$

✔ Answer: 4

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Problem 4:
$8 = 2(x - 5) + 6x$

Step 1: Distribute: $8 = 2x - 10 + 6x$

Step 2: Combine like terms: $8 = 8x - 10$

Step 3: Add 10: $18 = 8x$

Step 4: Divide by 8: $x = \frac{18}{8} = \frac{9}{4}$ or 2.25

Check: $2(\frac{9}{4} - 5) + 6(\frac{9}{4}) = 2(-\frac{11}{4}) + \frac{54}{4} = -\frac{22}{4} + \frac{54}{4} = \frac{32}{4} = 8 ✓$

✔ Answer: $\frac{9}{4}$

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Problem 5:
$-(x + 2) = 2(3x - 4)$

Step 1: Distribute both sides:

Left: $-x - 2$
Right: $6x - 8$

So: $-x - 2 = 6x - 8$

Step 2: Add x to both sides: $-2 = 7x - 8$

Step 3: Add 8: $6 = 7x$

Step 4: Divide by 7: $x = \frac{6}{7}$

Check: Left: $-(\frac{6}{7} + 2) = -\frac{20}{7}$
Right: $2(3 \cdot \frac{6}{7} - 4) = 2(\frac{18}{7} - \frac{28}{7}) = 2(-\frac{10}{7}) = -\frac{20}{7} ✓$

✔ Answer: $\frac{6}{7}$

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Problem 6:
$3 = 4(x - 2) + 5 - 3x$

Step 1: Distribute: $3 = 4x - 8 + 5 - 3x$

Step 2: Combine like terms: $3 = x - 3$

Step 3: Add 3: $6 = x$

Check: $4(6 - 2) + 5 - 3(6) = 4(4) + 5 - 18 = 16 + 5 - 18 = 3 ✓$

✔ Answer: 6

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Problem 7:
$\frac{2x - 1}{3} - \frac{3x}{4} = \frac{5}{6}$

Step 1: Common denominator for 3, 4, 6 → 12

Multiply entire equation by 12:

$12 \cdot \left( \frac{2x - 1}{3} \right) - 12 \cdot \left( \frac{3x}{4} \right) = 12 \cdot \left( \frac{5}{6} \right)$

→ $4(2x - 1) - 3(3x) = 10$

Step 2: Expand: $8x - 4 - 9x = 10$

Step 3: Combine: $-x - 4 = 10$

Step 4: Add 4: $-x = 14$

Step 5: Multiply by -1: $x = -14$

Check:
Left: $\frac{2(-14)-1}{3} - \frac{3(-14)}{4} = \frac{-29}{3} + \frac{42}{4} = \frac{-29}{3} + \frac{21}{2}$

Common denominator 6: $\frac{-58}{6} + \frac{63}{6} = \frac{5}{6} ✓$

✔ Answer: -14

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Problem 8:
$-5x + 3 = 2x + 8$

Step 1: Move all x to one side. Add 5x to both sides:

$3 = 7x + 8$

Step 2: Subtract 8: $-5 = 7x$

Step 3: Divide by 7: $x = -\frac{5}{7}$

Check:
Left: $-5(-\frac{5}{7}) + 3 = \frac{25}{7} + \frac{21}{7} = \frac{46}{7}$
Right: $2(-\frac{5}{7}) + 8 = -\frac{10}{7} + \frac{56}{7} = \frac{46}{7} ✓$

✔ Answer: $-\frac{5}{7}$

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Problem 9:
$4 = -(2x + 4)$

Step 1: Distribute the negative: $4 = -2x - 4$

Step 2: Add 4: $8 = -2x$

Step 3: Divide by -2: $x = -4$

Check: $-(2(-4) + 4) = -(-8 + 4) = -(-4) = 4 ✓$

✔ Answer: -4

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Problem 10:
$\frac{7}{8}y - 6 = 8$

Step 1: Add 6: $\frac{7}{8}y = 14$

Step 2: Multiply by reciprocal of $\frac{7}{8}$ → $\frac{8}{7}$

$y = 14 \cdot \frac{8}{7} = 2 \cdot 8 = 16$

Check: $\frac{7}{8}(16) - 6 = 14 - 6 = 8 ✓$

✔ Answer: 16

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Final Answers:

1. 25
2. 15
3. 4
4. $\frac{9}{4}$
5. $\frac{6}{7}$
6. 6
7. -14
8. $-\frac{5}{7}$
9. -4
10. 16