Multi-Step Equation Notes and Worksheets - Lindsay Bowden - Free Printable

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:
$$
-48 + 5 = 43x \\
-43 = 43x
$$

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:
$$
33 + 3 = -12x \\
36 = -12x
$$

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

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

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 (as implied):
Suppose someone wrote:
$$
\frac{1}{4} \cdot 12x - 16 = 2x \Rightarrow 3x - 16 = 2x
$$

✘ Mistake: They did not distribute $ \frac{1}{4} $ to both terms inside the parentheses. The subtraction is part of the expression, so you must multiply $-16$ by $ \frac{1}{4} $ as well.

Correct solution:
$$
\frac{1}{4}(12x - 16) = 2x \\
\frac{1}{4} \cdot 12x - \frac{1}{4} \cdot 16 = 2x \\
3x - 4 = 2x
$$

Now subtract $2x$ from both sides:
$$
3x - 2x - 4 = 0 \\
x - 4 = 0 \\
x = 4
$$

✔ Correct Answer: $ x = 4 $

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8. Find the mistake and correct it:

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

Suppose a student solved it like this:
$$
-4 + 14x = -12 + 12x \\
14x - 12x = -12 + 4 \\
2x = -8 \\
x = -4
$$

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

Alternatively, suppose the mistake was:

✘ Mistake Example:
Student adds $12$ to both sides first:
$$
-4 + 12 + 14x = 12x \\
8 + 14x = 12x
$$
Then subtracts $14x$:
$$
8 = -2x \Rightarrow x = -4
$$

This is still correct.

But perhaps the mistake is in incorrectly combining terms.

Let’s assume the student made this error:

> $-4 + 14x = -12 + 12x$
> Then they move $12x$ to the left and $-4$ to the right:
> $$
> 14x - 12x = -12 + 4 \\
> 2x = -8 \\
> x = -4
> $$

That’s correct.

But maybe the mistake is assuming no solution or infinite solutions incorrectly.

Wait — let's suppose the mistake is:

✘ Wrong:
They subtract $14x$ from both sides:
$$
-4 = -12 - 2x
$$
Then add 12:
$$
8 = -2x \Rightarrow x = -4
$$

Still correct.

So the only possible mistake might be not checking or misapplying operations.

But since the problem says “find the mistake,” and we're not given a worked-out solution, we assume a common error.

Let’s suppose the student did:

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

This is correct.

But perhaps the mistake is:

> Student thinks: $-4 + 14x = -12 + 12x$ → $14x - 12x = -12 + 4$ → $2x = -8$ → $x = -4$

But then plugs back in and says it's wrong?

Let’s verify:
Left: $-4 + 14(-4) = -4 - 56 = -60$
Right: $-12 + 12(-4) = -12 - 48 = -60$ ✔

It checks out.

So unless there's a specific incorrect solution shown, the most likely mistake is:

✘ Mistake: Not moving variables properly or mis-distributing.

But since no steps are shown, we can only assume a common error.

Let’s suppose the student added 12x to both sides instead of subtracting:

$$
-4 + 14x + 12x = -12 \\
-4 + 26x = -12
$$

Then:
$$
26x = -8 \Rightarrow x = -\frac{8}{26} = -\frac{4}{13}
$$

That would be wrong.

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

So the mistake could be: adding instead of subtracting $12x$.

✔ Correct Answer: $ x = -4 $

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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: Did not distribute $ \frac{1}{4} $ to both terms. Correct: $ x = 4 $ |
| 8 | Mistake: Possibly adding $12x$ instead of subtracting. Correct: $ x = -4 $ |

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