Printable worksheet for solving linear equations, featuring 14 problems across three sections with simplified fractions and decimals as answers.

The image you provided is a worksheet titled "Solving Linear Equations (D)" from Cazoom Maths Worksheets. It contains three sections (A, B, and C) with various linear equations to solve. 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: Remove parentheses, combine like terms, and simplify expressions.
2. Isolate the variable: Use inverse operations (addition/subtraction, multiplication/division) to isolate the variable on one side of the equation.
3. Solve for the variable: Perform any necessary arithmetic to find the value of the variable.
4. Check your solution: Substitute the solution back into the original equation to verify it satisfies the equation.

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Section A: Basic Linear Equations

#### Example 1: Solve $\frac{2x}{3} = 11$
1. Multiply both sides by 3 to eliminate the fraction:
$$
\frac{2x}{3} \cdot 3 = 11 \cdot 3
$$
$$
2x = 33
$$
2. Divide both sides by 2:
$$
x = \frac{33}{2}
$$
$$
x = 16.5
$$

#### Example 2: Solve $\frac{8 - 3x}{2} = 5$
1. Multiply both sides by 2 to eliminate the fraction:
$$
\frac{8 - 3x}{2} \cdot 2 = 5 \cdot 2
$$
$$
8 - 3x = 10
$$
2. Subtract 8 from both sides:
$$
-3x = 2
$$
3. Divide both sides by -3:
$$
x = \frac{2}{-3}
$$
$$
x = -\frac{2}{3}
$$

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Section B: Equations Involving Parentheses

#### Example 1: Solve $7(4 - 3x) - 8(2x + 5) = 0$
1. Distribute the constants:
$$
7 \cdot 4 - 7 \cdot 3x - 8 \cdot 2x - 8 \cdot 5 = 0
$$
$$
28 - 21x - 16x - 40 = 0
$$
2. Combine like terms:
$$
-21x - 16x + 28 - 40 = 0
$$
$$
-37x - 12 = 0
$$
3. Add 12 to both sides:
$$
-37x = 12
$$
4. Divide both sides by -37:
$$
x = \frac{12}{-37}
$$
$$
x = -\frac{12}{37}
$$

#### Example 2: Solve $2(4x - 5) = 5(2x - 5)$
1. Distribute the constants:
$$
2 \cdot 4x - 2 \cdot 5 = 5 \cdot 2x - 5 \cdot 5
$$
$$
8x - 10 = 10x - 25
$$
2. Subtract $8x$ from both sides:
$$
-10 = 2x - 25
$$
3. Add 25 to both sides:
$$
15 = 2x
$$
4. Divide both sides by 2:
$$
x = \frac{15}{2}
$$
$$
x = 7.5
$$

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Section C: Complex Fractions and Multi-Step Equations

#### Example 1: Solve $\frac{3x - 2}{4} = \frac{x + 1}{2}$
1. Eliminate the fractions by finding a common denominator (in this case, 4):
$$
\frac{3x - 2}{4} = \frac{2(x + 1)}{4}
$$
$$
3x - 2 = 2(x + 1)
$$
2. Distribute on the right-hand side:
$$
3x - 2 = 2x + 2
$$
3. Subtract $2x$ from both sides:
$$
x - 2 = 2
$$
4. Add 2 to both sides:
$$
x = 4
$$

#### Example 2: Solve $\frac{5(3x + 1)}{4} = \frac{2(1 + x)}{3}$
1. Eliminate the fractions by cross-multiplying:
$$
5(3x + 1) \cdot 3 = 4 \cdot 2(1 + x)
$$
$$
15(3x + 1) = 8(1 + x)
$$
2. Distribute on both sides:
$$
45x + 15 = 8 + 8x
$$
3. Subtract $8x$ from both sides:
$$
45x - 8x + 15 = 8
$$
$$
37x + 15 = 8
$$
4. Subtract 15 from both sides:
$$
37x = -7
$$
5. Divide both sides by 37:
$$
x = \frac{-7}{37}
$$

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Final Answer

The solutions to the problems in the worksheet can be found by applying the steps outlined above. For brevity, I have demonstrated the process with selected examples. If you need solutions to specific problems, please let me know!

$$
\boxed{\text{Solutions are derived using the methods shown above.}}
$$