Comprehensive practice sheet featuring eight multi-step equation problems, including distribution, fractions, and error analysis tasks.

Let's solve each of these multi-step equations step by step, showing all work. We'll also identify and correct the mistakes in problems 7 and 8.

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1. $ 4(3x + 7) = 64 + 4x $

Step 1: Distribute the 4 on the left side:
$$
4 \cdot 3x + 4 \cdot 7 = 64 + 4x \\
12x + 28 = 64 + 4x
$$

Step 2: Subtract $4x$ from both sides:
$$
12x - 4x + 28 = 64 \\
8x + 28 = 64
$$

Step 3: Subtract 28 from both sides:
$$
8x = 64 - 28 \\
8x = 36
$$

Step 4: Divide by 8:
$$
x = \frac{36}{8} = \frac{9}{2} = 4.5
$$

✔ Answer: $ x = \frac{9}{2} $ or $ 4.5 $

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2. $ 16 = -4(2x - 5) $

Step 1: Distribute the $-4$:
$$
16 = -4 \cdot 2x + (-4) \cdot (-5) \\
16 = -8x + 20
$$

Step 2: Subtract 20 from both sides:
$$
16 - 20 = -8x \\
-4 = -8x
$$

Step 3: Divide by $-8$:
$$
x = \frac{-4}{-8} = \frac{1}{2}
$$

✔ Answer: $ x = \frac{1}{2} $

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3. $ -8(6 + 5x) = 3x - 5 $

Step 1: Distribute $-8$:
$$
-8 \cdot 6 + (-8) \cdot 5x = 3x - 5 \\
-48 - 40x = 3x - 5
$$

Step 2: Add $40x$ to both sides:
$$
-48 = 43x - 5
$$

Step 3: Add 5 to both sides:
$$
-43 = 43x
$$

Step 4: Divide by 43:
$$
x = \frac{-43}{43} = -1
$$

✔ Answer: $ x = -1 $

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4. $ 33 = -3(1 + 4x) $

Step 1: Distribute $-3$:
$$
33 = -3 \cdot 1 + (-3) \cdot 4x \\
33 = -3 - 12x
$$

Step 2: Add 3 to both sides:
$$
36 = -12x
$$

Step 3: Divide by $-12$:
$$
x = \frac{36}{-12} = -3
$$

✔ Answer: $ x = -3 $

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5. $ \frac{4x - 2}{5} = 0.4 $

Step 1: Multiply both sides by 5 to eliminate denominator:
$$
4x - 2 = 5 \cdot 0.4 \\
4x - 2 = 2
$$

Step 2: Add 2 to both sides:
$$
4x = 4
$$

Step 3: Divide by 4:
$$
x = 1
$$

✔ Answer: $ x = 1 $

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6. $ \frac{2}{3}x - 6 = \frac{1}{3}x + 3 $

Step 1: Subtract $ \frac{1}{3}x $ from both sides:
$$
\left(\frac{2}{3}x - \frac{1}{3}x\right) - 6 = 3 \\
\frac{1}{3}x - 6 = 3
$$

Step 2: Add 6 to both sides:
$$
\frac{1}{3}x = 9
$$

Step 3: Multiply both sides by 3:
$$
x = 27
$$

✔ Answer: $ x = 27 $

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7. Find the mistake and correct it:

$$
\frac{1}{4}(12x - 16) = 2x
$$

Incorrect solution (assumed):
Suppose someone did this:
$$
\frac{1}{4} \cdot 12x - \frac{1}{4} \cdot 16 = 2x \\
3x - 4 = 2x
$$
Then subtract $2x$:
$$
x - 4 = 0 \Rightarrow x = 4
$$

Wait — that’s actually correct, but let’s check if there’s a common mistake.

But perhaps the mistake is assuming that distributing is optional or incorrectly applied.

Let’s do it correctly:

Correct steps:
$$
\frac{1}{4}(12x - 16) = 2x
$$

Distribute:
$$
\frac{1}{4} \cdot 12x = 3x, \quad \frac{1}{4} \cdot (-16) = -4 \\
3x - 4 = 2x
$$

Subtract $2x$:
$$
x - 4 = 0 \Rightarrow x = 4
$$

✔ So the correct answer is $ x = 4 $.

But maybe the mistake was not distributing properly? For example, someone might have done:
$$
\frac{1}{4}(12x - 16) = 2x \Rightarrow \frac{1}{4} \cdot 12x - 16 = 2x \quad \text{(✘ Wrong!)}
$$
This is incorrect because you must distribute $ \frac{1}{4} $ to both terms.

So the mistake is: Failing to distribute $ \frac{1}{4} $ to both terms inside the parentheses.

✔ Correction:
$$
\frac{1}{4}(12x - 16) = 2x \\
3x - 4 = 2x \\
x = 4
$$

✔️ Final answer: $ x = 4 $

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8. Find the mistake and correct it:

$$
-4 + 14x = -12 + 12x
$$

Common mistake: Incorrectly moving terms without proper sign changes.

Let’s suppose someone did:
$$
-4 + 14x = -12 + 12x \\
14x - 12x = -12 + 4 \\
2x = -8 \\
x = -4
$$

Wait — that’s actually correct.

But maybe the mistake is in sign error during subtraction.

Let’s do it step-by-step:

Correct solution:

Start:
$$
-4 + 14x = -12 + 12x
$$

Step 1: Subtract $12x$ from both sides:
$$
-4 + 2x = -12
$$

Step 2: Add 4 to both sides:
$$
2x = -8
$$

Step 3: Divide by 2:
$$
x = -4
$$

✔ So the correct answer is $ x = -4 $.

But what could be the mistake?

Possibility: Someone might have written:
$$
-4 + 14x = -12 + 12x \\
14x - 12x = -12 + 4 \quad \text{(this is correct)} \\
2x = -8 \Rightarrow x = -4
$$

But maybe they did:
$$
14x - 12x = -12 - 4 \quad \text{(✘ Mistake!)} \Rightarrow 2x = -16 \Rightarrow x = -8
$$

So the mistake is: Adding instead of subtracting when moving constants.

Specifically, when moving $-4$ to the right, it should become $+4$, not $-4$. But here, the error is likely:

> Mistake: When moving $-12$ to the left, it becomes $+12$, but if someone does $-12 + 4 = -8$, that’s correct. But if they mistakenly do $-12 - 4 = -16$, then $x = -8$.

So the mistake is: Incorrectly combining constants with wrong signs.

✔ Corrected version:
$$
-4 + 14x = -12 + 12x \\
\text{Add } 12 \text{ to both sides: } 8 + 14x = 12x \\
\text{Subtract } 12x: 8 + 2x = 0 \\
\text{Subtract 8: } 2x = -8 \\
x = -4
$$

Or better:
$$
-4 + 14x = -12 + 12x \\
\text{Subtract } 12x: -4 + 2x = -12 \\
\text{Add 4: } 2x = -8 \\
x = -4
$$

✔️ Final answer: $ x = -4 $

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✔ Summary of Answers:

| Problem | Answer |
|--------|--------|
| 1 | $ x = \frac{9}{2} $ |
| 2 | $ x = \frac{1}{2} $ |
| 3 | $ x = -1 $ |
| 4 | $ x = -3 $ |
| 5 | $ x = 1 $ |
| 6 | $ x = 27 $ |
| 7 | Mistake: Not distributing $ \frac{1}{4} $ to both terms; Correct: $ x = 4 $ |
| 8 | Mistake: Incorrect sign when moving terms (e.g., $-12 - 4$ instead of $-12 + 4$); Correct: $ x = -4 $ |

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