Multi-step equation practice worksheet for algebra students.

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:
$$
4 \cdot 3x + 4 \cdot 7 = 12x + 28
$$
So:
$$
12x + 28 = 64 + 4x
$$

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

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

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 $-4$:
$$
16 = -8x + 20
$$

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

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

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 $-3$:
$$
33 = -3 - 12x
$$

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:
$$
4x - 2 = 0.4 \cdot 5 = 2
$$

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:
$$
\left(\frac{2}{3}x - \frac{1}{3}x\right) - 6 = 3 \Rightarrow \frac{1}{3}x - 6 = 3
$$

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:

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

Mistake: The problem says "find the mistake and correct it", but no solution is shown. So we’ll solve it correctly and point out what a common mistake might be.

Correct Solution:

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

Subtract $2x$ from both sides:
$$
x - 4 = 0 \Rightarrow x = 4
$$

✔ Correct Answer: $ x = 4 $

Common Mistake: A student might forget to distribute the $ \frac{1}{4} $ to both terms inside the parentheses and instead write:
$$
\frac{1}{4}(12x - 16) = \frac{1}{4} \cdot 12x = 3x \quad \text{(and ignore the } -16\text{)}
$$
Then incorrectly say $ 3x = 2x $, leading to $ x = 0 $. That’s wrong.

✘ Incorrect Step: $ \frac{1}{4}(12x - 16) = 3x $ (missing $-4$)

✔ Correction: Always distribute to both terms: $ 3x - 4 = 2x $

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8. Find the mistake and correct it:

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

Again, no steps are shown, so we solve it correctly and identify possible errors.

Correct Solution:

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

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

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

✔ Correct Answer: $ x = -4 $

Common Mistake: A student might subtract $14x$ from both sides instead:
$$
-4 = -12 - 2x \quad \text{(wrong direction)}
$$
Or maybe combine constants incorrectly.

Another possible error: adding $12x$ to both sides instead of subtracting:
$$
-4 + 14x + 12x = -12 \Rightarrow -4 + 26x = -12 \quad \text{(incorrect)}
$$

But if someone wrote:
$$
-4 + 14x = -12 + 12x \\
\Rightarrow 14x - 12x = -12 + 4 \\
\Rightarrow 2x = -8 \Rightarrow x = -4
$$
That’s correct.

So perhaps the original "mistake" was not shown — but assuming a student made an error like:

✘ Incorrect: $ 14x - 12x = -12 + 4 \Rightarrow 2x = -8 \Rightarrow x = 4 $ → sign error

✔ Correction: $ 2x = -8 \Rightarrow x = -4 $

Or maybe they moved terms without changing signs.

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✔ Final Answers Summary:

| 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: Sign error when moving terms; Correct: $ x = -4 $ |

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