Worksheet for solving linear equations with simplified fraction and decimal answers.

The image you provided is a worksheet titled "Solving Linear Equations (D)" from the Corbettmaths educational resources. The task involves solving linear equations and expressing the answers as simplified fractions or decimals. Below, I will outline the general approach to solving these types of problems and provide examples for each section.

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General Approach to Solving Linear Equations

1. Simplify both sides: Expand any brackets and combine like terms.
2. Isolate the variable: Move all terms involving the variable to one side of the equation and all constant terms to the other side.
3. Solve for the variable: Divide by the coefficient of the variable to isolate it.
4. Simplify the answer: Express the solution as a fraction or decimal, depending on the instructions.

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Section A: Basic Linear Equations

#### Example 1: Solve \( \frac{2x}{3} - 11 = -5 \)
1. Add 11 to both sides:
\[
\frac{2x}{3} = 6
\]
2. Multiply both sides by 3 to eliminate the fraction:
\[
2x = 18
\]
3. Divide by 2:
\[
x = 9
\]

#### Example 2: Solve \( 2 - \frac{3x}{5} = -2 \)
1. Subtract 2 from both sides:
\[
-\frac{3x}{5} = -4
\]
2. Multiply both sides by -5:
\[
3x = 20
\]
3. Divide by 3:
\[
x = \frac{20}{3}
\]

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Section B: Equations with Brackets

#### Example 7: Solve \( 4(2x - 3) - 3(2x + 5) = 9 \)
1. Expand the brackets:
\[
8x - 12 - 6x - 15 = 9
\]
2. Combine like terms:
\[
2x - 27 = 9
\]
3. Add 27 to both sides:
\[
2x = 36
\]
4. Divide by 2:
\[
x = 18
\]

#### Example 10: Solve \( 12(x + 2) - 3(2x + 6) = 9(x - 3) \)
1. Expand the brackets:
\[
12x + 24 - 6x - 18 = 9x - 27
\]
2. Combine like terms:
\[
6x + 6 = 9x - 27
\]
3. Subtract 6x from both sides:
\[
6 = 3x - 27
\]
4. Add 27 to both sides:
\[
33 = 3x
\]
5. Divide by 3:
\[
x = 11
\]

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Section C: Complex Fractions

#### Example 3: Solve \( \frac{7x - 8}{2x - 5} = \frac{3}{2} \)
1. Cross-multiply:
\[
2(7x - 8) = 3(2x - 5)
\]
2. Expand both sides:
\[
14x - 16 = 6x - 15
\]
3. Subtract 6x from both sides:
\[
8x - 16 = -15
\]
4. Add 16 to both sides:
\[
8x = 1
\]
5. Divide by 8:
\[
x = \frac{1}{8}
\]

#### Example 12: Solve \( \frac{x + 4}{3x + 10} = \frac{1}{2} \)
1. Cross-multiply:
\[
2(x + 4) = 1(3x + 10)
\]
2. Expand both sides:
\[
2x + 8 = 3x + 10
\]
3. Subtract 2x from both sides:
\[
8 = x + 10
\]
4. Subtract 10 from both sides:
\[
x = -2
\]

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Final Answers

Here are the solutions for the first few problems in each section as examples:

- Section A:
1. \( x = 9 \)
2. \( x = \frac{20}{3} \)

- Section B:
7. \( x = 18 \)
10. \( x = 11 \)

- Section C:
3. \( x = \frac{1}{8} \)
12. \( x = -2 \)

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If you need solutions for specific problems or further clarification, feel free to ask!

Final Answer:
\[
\boxed{x = 9, \, x = \frac{20}{3}, \, x = 18, \, x = 11, \, x = \frac{1}{8}, \, x = -2}
\]