Linear Equations Worksheet from Math Monks, designed to help students practice solving equations with variables on both sides and fractional coefficients.

Linear Equations Worksheet Solution

We will solve each equation step by step to find the unknown variable.

---

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

1. Multiply both sides by $ -\frac{5}{9} $ to isolate $ x $:
\[
x = -45 \cdot \left(-\frac{5}{9}\right)
\]
2. Simplify:
\[
x = \frac{45 \cdot 5}{9} = \frac{225}{9} = 25
\]

Solution: $ x = 25 $

---

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

1. Find a common denominator for the fractions, which is 15:
\[
\frac{x}{3} = \frac{5x}{15}, \quad \frac{x}{5} = \frac{3x}{15}
\]
2. Rewrite the equation:
\[
\frac{5x}{15} - \frac{3x}{15} = 2
\]
3. Combine the fractions:
\[
\frac{5x - 3x}{15} = 2 \implies \frac{2x}{15} = 2
\]
4. Multiply both sides by 15:
\[
2x = 30
\]
5. Divide by 2:
\[
x = 15
\]

Solution: $ x = 15 $

---

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

1. Eliminate the denominator by multiplying both sides by 6:
\[
4x + 5 = \frac{7}{2} \cdot 6
\]
2. Simplify the right-hand side:
\[
4x + 5 = 21
\]
3. Subtract 5 from both sides:
\[
4x = 16
\]
4. Divide by 4:
\[
x = 4
\]

Solution: $ x = 4 $

---

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

1. Distribute the 2 on the right-hand side:
\[
8 = 2x - 10 + 6x
\]
2. Combine like terms:
\[
8 = 8x - 10
\]
3. Add 10 to both sides:
\[
18 = 8x
\]
4. Divide by 8:
\[
x = \frac{18}{8} = \frac{9}{4}
\]

Solution: $ x = \frac{9}{4} $

---

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

1. Distribute the negative sign on the left and the 2 on the right:
\[
-x - 2 = 6x - 8
\]
2. Add $ x $ to both sides:
\[
-2 = 7x - 8
\]
3. Add 8 to both sides:
\[
6 = 7x
\]
4. Divide by 7:
\[
x = \frac{6}{7}
\]

Solution: $ x = \frac{6}{7} $

---

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

1. Distribute the 4 on the right-hand side:
\[
3 = 4x - 8 + 5 - 3x
\]
2. Combine like terms:
\[
3 = (4x - 3x) + (-8 + 5) \implies 3 = x - 3
\]
3. Add 3 to both sides:
\[
6 = x
\]

Solution: $ x = 6 $

---

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

1. Find a common denominator for the fractions, which is 12:
\[
\frac{2x - 1}{3} = \frac{4(2x - 1)}{12} = \frac{8x - 4}{12}, \quad \frac{3x}{4} = \frac{9x}{12}
\]
2. Rewrite the equation:
\[
\frac{8x - 4}{12} - \frac{9x}{12} = \frac{5}{6}
\]
3. Combine the fractions on the left-hand side:
\[
\frac{8x - 4 - 9x}{12} = \frac{5}{6} \implies \frac{-x - 4}{12} = \frac{5}{6}
\]
4. Eliminate the denominators by multiplying both sides by 12:
\[
-x - 4 = \frac{5}{6} \cdot 12 \implies -x - 4 = 10
\]
5. Add 4 to both sides:
\[
-x = 14
\]
6. Multiply by -1:
\[
x = -14
\]

Solution: $ x = -14 $

---

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

1. Add $ 5x $ to both sides:
\[
3 = 7x + 8
\]
2. Subtract 8 from both sides:
\[
-5 = 7x
\]
3. Divide by 7:
\[
x = -\frac{5}{7}
\]

Solution: $ x = -\frac{5}{7} $

---

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

1. Distribute the negative sign on the right-hand side:
\[
4 = -2x - 4
\]
2. Add 4 to both sides:
\[
8 = -2x
\]
3. Divide by -2:
\[
x = -4
\]

Solution: $ x = -4 $

---

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

1. Add 6 to both sides:
\[
\frac{7}{8}y = 14
\]
2. Multiply both sides by $ \frac{8}{7} $ to isolate $ y $:
\[
y = 14 \cdot \frac{8}{7} = 2 \cdot 8 = 16
\]

Solution: $ y = 16 $

---

Final Answers:

\[
\boxed{
\begin{aligned}
1. & \ x = 25 \\
2. & \ x = 15 \\
3. & \ x = 4 \\
4. & \ x = \frac{9}{4} \\
5. & \ x = \frac{6}{7} \\
6. & \ x = 6 \\
7. & \ x = -14 \\
8. & \ x = -\frac{5}{7} \\
9. & \ x = -4 \\
10. & \ y = 16
\end{aligned}
}
\]