Linear Equations Worksheets with Answer Key - Free Printable

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:

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