Math worksheet for practicing solving multi-step equations with space to show work.

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

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

---

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

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

Step 2: Add $12x$ to both sides:
$$
33 + 12x = -3
$$

Step 3: Subtract 33 from both sides:
$$
12x = -3 - 33 = -36
$$

Step 4: 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:
$$
4x = 2 + 2 = 4
$$

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

✔ 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:
$$
\frac{1}{3}x = 3 + 6 = 9
$$

Step 3: Multiply both sides by 3:
$$
x = 9 \cdot 3 = 27
$$

✔ Answer: $ x = 27 $

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7. Find the mistake and correct it:

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

Incorrect solution (assumed):
If someone incorrectly distributes:
$$
\frac{1}{4} \cdot 12x = 3x,\quad \frac{1}{4} \cdot (-16) = -4
\Rightarrow 3x - 4 = 2x
$$
Then subtract $2x$: $x - 4 = 0$, so $x = 4$

But wait — that’s actually correct! Let's check:

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

Now verify:
Left: $ \frac{1}{4}(12(4) - 16) = \frac{1}{4}(48 - 16) = \frac{32}{4} = 8 $
Right: $ 2(4) = 8 $

✔ So the equation is correct, but maybe the student didn't show steps?

Wait — the problem says “Find the mistake and correct it.” But there's no visible mistake in the equation itself. Perhaps a common mistake is not distributing correctly, such as:

✘ Mistake: $ \frac{1}{4}(12x - 16) = 3x - 16 $ → wrong!

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

So mistake: Forgetting to multiply $-16$ by $\frac{1}{4}$

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

Answer: The mistake is not distributing the $\frac{1}{4}$ to both terms. Correct distribution gives $3x - 4 = 2x$, so $x = 4$.

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8. Find the mistake and correct it:

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

Suppose someone does this:

Add $12$ to both sides: $8 + 14x = 12x$

Subtract $14x$: $8 = -2x$, so $x = -4$

But let's do it properly:

Step 1: Subtract $12x$ from both sides:
$$
-4 + 14x - 12x = -12 \Rightarrow -4 + 2x = -12
$$

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

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

So again, if they did it correctly, they get $x = -4$. But maybe the mistake is in combining like terms incorrectly.

For example, suppose someone writes:

✘ Mistake: $-4 + 14x = -12 + 12x$ → subtract $12x$: $-4 + 2x = -12$ → then adds $4$ to get $2x = -8$, $x = -4$ — that's correct.

But maybe someone adds $12x$ to both sides instead of subtracting, or misplaces signs.

Another possible mistake:
Someone might write:
$$
-4 + 14x = -12 + 12x \\
\Rightarrow 14x - 12x = -12 + 4 \Rightarrow 2x = -8 \Rightarrow x = -4
$$

That’s correct.

So perhaps the mistake is in assuming no solution or infinite solutions when it has one.

Or maybe a common error:
✘ Incorrect: Move $12x$ to left as $+12x$: $-4 + 14x + 12x = -12$ → $-4 + 26x = -12$ → $26x = -8$ → $x = -\frac{4}{13}$ — wrong!

This is incorrect because you should subtract $12x$, not add.

✔ Correct:
$$
-4 + 14x = -12 + 12x \\
\text{Subtract } 12x: \quad -4 + 2x = -12 \\
\text{Add } 4: \quad 2x = -8 \\
\text{Divide: } x = -4
$$

Mistake: Adding $12x$ instead of subtracting.
Correction: Subtract $12x$ from both sides.

✔ 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: $3x - 4 = 2x$, so $x = 4$ |
| 8 | Mistake: Adding $12x$ instead of subtracting. Correct: $-4 + 2x = -12$, so $x = -4$ |

Let me know if you'd like these written out neatly for a worksheet!