Linear Equations Worksheets with Answer Key - Free Printable

Problem: Solve the given linear equations.

We will solve each equation step by step. Here are the equations:

1. \( x + 6 = 9 \)
2. \( 3(x - 2) = 6 \)
3. \( 10 - x = 6 \)
4. \( -y = -12 \)
5. \( 6 - \frac{x}{4} = -\frac{x}{4} \)
6. \( -5x = -7 + 2x \)
7. \( n - 4n = 14 - 3n \)
8. \( \frac{1}{7}x = 2 \)
9. \( 4x = -32 \)
10. \( 4 = -2(x + 3) \)
11. \( \frac{x}{3} = -9 \)
12. \( -\frac{9}{6}x = -40 \)
13. \( -2x - 6x = 1 \)
14. \( 10x - x = -9 \)

---

Solution:

#### 1. \( x + 6 = 9 \)
- Subtract 6 from both sides:
\[
x + 6 - 6 = 9 - 6
\]
\[
x = 3
\]

#### 2. \( 3(x - 2) = 6 \)
- Distribute the 3 on the left side:
\[
3x - 6 = 6
\]
- Add 6 to both sides:
\[
3x - 6 + 6 = 6 + 6
\]
\[
3x = 12
\]
- Divide both sides by 3:
\[
x = 4
\]

#### 3. \( 10 - x = 6 \)
- Subtract 10 from both sides:
\[
10 - x - 10 = 6 - 10
\]
\[
-x = -4
\]
- Multiply both sides by -1:
\[
x = 4
\]

#### 4. \( -y = -12 \)
- Multiply both sides by -1:
\[
y = 12
\]

#### 5. \( 6 - \frac{x}{4} = -\frac{x}{4} \)
- Add \(\frac{x}{4}\) to both sides:
\[
6 - \frac{x}{4} + \frac{x}{4} = -\frac{x}{4} + \frac{x}{4}
\]
\[
6 = 0
\]
- This equation has no solution because it leads to a contradiction.

#### 6. \( -5x = -7 + 2x \)
- Subtract \(2x\) from both sides:
\[
-5x - 2x = -7 + 2x - 2x
\]
\[
-7x = -7
\]
- Divide both sides by -7:
\[
x = 1
\]

#### 7. \( n - 4n = 14 - 3n \)
- Simplify the left side:
\[
-3n = 14 - 3n
\]
- Add \(3n\) to both sides:
\[
-3n + 3n = 14 - 3n + 3n
\]
\[
0 = 14
\]
- This equation has no solution because it leads to a contradiction.

#### 8. \( \frac{1}{7}x = 2 \)
- Multiply both sides by 7:
\[
7 \cdot \frac{1}{7}x = 2 \cdot 7
\]
\[
x = 14
\]

#### 9. \( 4x = -32 \)
- Divide both sides by 4:
\[
x = \frac{-32}{4}
\]
\[
x = -8
\]

#### 10. \( 4 = -2(x + 3) \)
- Distribute the -2 on the right side:
\[
4 = -2x - 6
\]
- Add 6 to both sides:
\[
4 + 6 = -2x - 6 + 6
\]
\[
10 = -2x
\]
- Divide both sides by -2:
\[
x = \frac{10}{-2}
\]
\[
x = -5
\]

#### 11. \( \frac{x}{3} = -9 \)
- Multiply both sides by 3:
\[
3 \cdot \frac{x}{3} = -9 \cdot 3
\]
\[
x = -27
\]

#### 12. \( -\frac{9}{6}x = -40 \)
- Simplify \(-\frac{9}{6}\):
\[
-\frac{9}{6} = -\frac{3}{2}
\]
- The equation becomes:
\[
-\frac{3}{2}x = -40
\]
- Multiply both sides by \(-\frac{2}{3}\):
\[
x = -40 \cdot -\frac{2}{3}
\]
\[
x = \frac{80}{3}
\]

#### 13. \( -2x - 6x = 1 \)
- Combine like terms on the left side:
\[
-8x = 1
\]
- Divide both sides by -8:
\[
x = \frac{1}{-8}
\]
\[
x = -\frac{1}{8}
\]

#### 14. \( 10x - x = -9 \)
- Combine like terms on the left side:
\[
9x = -9
\]
- Divide both sides by 9:
\[
x = \frac{-9}{9}
\]
\[
x = -1
\]

---

Final Answers:

\[
\boxed{
\begin{aligned}
1. & \ x = 3 \\
2. & \ x = 4 \\
3. & \ x = 4 \\
4. & \ y = 12 \\
5. & \ \text{No solution} \\
6. & \ x = 1 \\
7. & \ \text{No solution} \\
8. & \ x = 14 \\
9. & \ x = -8 \\
10. & \ x = -5 \\
11. & \ x = -27 \\
12. & \ x = \frac{80}{3} \\
13. & \ x = -\frac{1}{8} \\
14. & \ x = -1
\end{aligned}
}
\]