Solving Linear Equations with the Distributive Property Worksheet ... - Free Printable

Problem: Solving Linear Equations with the Distributive Property

The task involves solving linear equations by expanding brackets and simplifying. Let's solve each equation step by step.

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#### Section A: Expand the brackets and solve the following equations

1. Equation: \( 3(x + 6) = 27 \)
- Expand the brackets:
\[
3x + 18 = 27
\]
- Subtract 18 from both sides:
\[
3x = 9
\]
- Divide by 3:
\[
x = 3
\]

2. Equation: \( 2(x + 4) = 22 \)
- Expand the brackets:
\[
2x + 8 = 22
\]
- Subtract 8 from both sides:
\[
2x = 14
\]
- Divide by 2:
\[
x = 7
\]

3. Equation: \( 6(x - 5) = 42 \)
- Expand the brackets:
\[
6x - 30 = 42
\]
- Add 30 to both sides:
\[
6x = 72
\]
- Divide by 6:
\[
x = 12
\]

4. Equation: \( 2(x - 3) = 16 \)
- Expand the brackets:
\[
2x - 6 = 16
\]
- Add 6 to both sides:
\[
2x = 22
\]
- Divide by 2:
\[
x = 11
\]

5. Equation: \( 4(2x + 5) = 36 \)
- Expand the brackets:
\[
8x + 20 = 36
\]
- Subtract 20 from both sides:
\[
8x = 16
\]
- Divide by 8:
\[
x = 2
\]

6. Equation: \( 8(3x - 2) = 80 \)
- Expand the brackets:
\[
24x - 16 = 80
\]
- Add 16 to both sides:
\[
24x = 96
\]
- Divide by 24:
\[
x = 4
\]

7. Equation: \( 3(6x - 7) = 15 \)
- Expand the brackets:
\[
18x - 21 = 15
\]
- Add 21 to both sides:
\[
18x = 36
\]
- Divide by 18:
\[
x = 2
\]

8. Equation: \( 9(7x + 1) = 72 \)
- Expand the brackets:
\[
63x + 9 = 72
\]
- Subtract 9 from both sides:
\[
63x = 63
\]
- Divide by 63:
\[
x = 1
\]

9. Equation: \( 4(3x + 5) = 8 \)
- Expand the brackets:
\[
12x + 20 = 8
\]
- Subtract 20 from both sides:
\[
12x = -12
\]
- Divide by 12:
\[
x = -1
\]

10. Equation: \( 2(4x + 2) = -12 \)
- Expand the brackets:
\[
8x + 4 = -12
\]
- Subtract 4 from both sides:
\[
8x = -16
\]
- Divide by 8:
\[
x = -2
\]

11. Equation: \( 10(3x - 1) = -100 \)
- Expand the brackets:
\[
30x - 10 = -100
\]
- Add 10 to both sides:
\[
30x = -90
\]
- Divide by 30:
\[
x = -3
\]

12. Equation: \( 12(2x + 7) = 96 \)
- Expand the brackets:
\[
24x + 84 = 96
\]
- Subtract 84 from both sides:
\[
24x = 12
\]
- Divide by 24:
\[
x = \frac{1}{2}
\]

---

#### Section B: Solve the following equations

1. Equation: \( \frac{x + 8}{2} = 9 \)
- Multiply both sides by 2:
\[
x + 8 = 18
\]
- Subtract 8 from both sides:
\[
x = 10
\]

2. Equation: \( \frac{x + 6}{5} = 7 \)
- Multiply both sides by 5:
\[
x + 6 = 35
\]
- Subtract 6 from both sides:
\[
x = 29
\]

3. Equation: \( \frac{2x + 5}{3} = 9 \)
- Multiply both sides by 3:
\[
2x + 5 = 27
\]
- Subtract 5 from both sides:
\[
2x = 22
\]
- Divide by 2:
\[
x = 11
\]

4. Equation: \( \frac{6(7x + 2)}{9} = 6 \)
- Simplify the fraction:
\[
\frac{2(7x + 2)}{3} = 6
\]
- Multiply both sides by 3:
\[
2(7x + 2) = 18
\]
- Expand the brackets:
\[
14x + 4 = 18
\]
- Subtract 4 from both sides:
\[
14x = 14
\]
- Divide by 14:
\[
x = 1
\]

5. Equation: \( \frac{2(3x + 4)}{7} = -4 \)
- Multiply both sides by 7:
\[
2(3x + 4) = -28
\]
- Expand the brackets:
\[
6x + 8 = -28
\]
- Subtract 8 from both sides:
\[
6x = -36
\]
- Divide by 6:
\[
x = -6
\]

6. Equation: \( \frac{3(4x - 6)}{9} = -6 \)
- Simplify the fraction:
\[
\frac{4x - 6}{3} = -6
\]
- Multiply both sides by 3:
\[
4x - 6 = -18
\]
- Add 6 to both sides:
\[
4x = -12
\]
- Divide by 4:
\[
x = -3
\]

7. Equation: \( \frac{1}{2}(4x + 6) = 9 \)
- Multiply both sides by 2:
\[
4x + 6 = 18
\]
- Subtract 6 from both sides:
\[
4x = 12
\]
- Divide by 4:
\[
x = 3
\]

8. Equation: \( \frac{1}{2}(2x + 12) = 1 \)
- Multiply both sides by 2:
\[
2x + 12 = 2
\]
- Subtract 12 from both sides:
\[
2x = -10
\]
- Divide by 2:
\[
x = -5
\]

9. Equation: \( \frac{1}{2}(3x + 5) = 13 \)
- Multiply both sides by 2:
\[
3x + 5 = 26
\]
- Subtract 5 from both sides:
\[
3x = 21
\]
- Divide by 3:
\[
x = 7
\]

---

Final Answers:

\[
\boxed{
\begin{array}{ccc}
\text{Section A:} & & \\
1. & x = 3 & 5. & x = 2 & 9. & x = -1 \\
2. & x = 7 & 6. & x = 4 & 10. & x = -2 \\
3. & x = 12 & 7. & x = 2 & 11. & x = -3 \\
4. & x = 11 & 8. & x = 1 & 12. & x = \frac{1}{2} \\
\\
\text{Section B:} & & \\
1. & x = 10 & 4. & x = 1 & 7. & x = 3 \\
2. & x = 29 & 5. & x = -6 & 8. & x = -5 \\
3. & x = 11 & 6. & x = -3 & 9. & x = 7 \\
\end{array}
}
\]