Solving Multi-Step Equations Variables on Both Sides Notes ... - Free Printable

Let's solve each of these multi-step equations step by step, showing all work and checking the answers.

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1) $ 4 - 8m + 1 = 11 - 2m $

Step 1: Combine like terms on the left side:
$$
(4 + 1) - 8m = 11 - 2m \Rightarrow 5 - 8m = 11 - 2m
$$

Step 2: Add $ 8m $ to both sides to get variables on one side:
$$
5 = 11 + 6m
$$

Step 3: Subtract 11 from both sides:
$$
5 - 11 = 6m \Rightarrow -6 = 6m
$$

Step 4: Divide by 6:
$$
m = -1
$$

✔ Check:
Left: $ 4 - 8(-1) + 1 = 4 + 8 + 1 = 13 $
Right: $ 11 - 2(-1) = 11 + 2 = 13 $
✔️ Both sides equal → Correct

---

2) $ -4x + 1 + 9x = 8 - 3x $

Step 1: Combine like terms on the left:
$$
(-4x + 9x) + 1 = 8 - 3x \Rightarrow 5x + 1 = 8 - 3x
$$

Step 2: Add $ 3x $ to both sides:
$$
8x + 1 = 8
$$

Step 3: Subtract 1:
$$
8x = 7
$$

Step 4: Divide by 8:
$$
x = \frac{7}{8}
$$

✔ Check:
Left: $ -4(\frac{7}{8}) + 1 + 9(\frac{7}{8}) = -\frac{28}{8} + 1 + \frac{63}{8} = (-3.5 + 1 + 7.875) = 5.375 $
Right: $ 8 - 3(\frac{7}{8}) = 8 - \frac{21}{8} = 8 - 2.625 = 5.375 $
✔️ Both sides equal → Correct

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3) $ 1 + \frac{1}{4}y = 2y - 2y + 5 $

Step 1: Simplify right side:
$$
2y - 2y = 0 \Rightarrow 1 + \frac{1}{4}y = 5
$$

Step 2: Subtract 1:
$$
\frac{1}{4}y = 4
$$

Step 3: Multiply both sides by 4:
$$
y = 16
$$

✔ Check:
Left: $ 1 + \frac{1}{4}(16) = 1 + 4 = 5 $
Right: $ 2(16) - 2(16) + 5 = 0 + 5 = 5 $
✔️ Correct

---

4) $ 25 + \frac{1}{3}y = -\frac{1}{3}y - 8 + \frac{2}{3}y $

Step 1: Combine like terms on the right:
$$
-\frac{1}{3}y + \frac{2}{3}y = \frac{1}{3}y \Rightarrow 25 + \frac{1}{3}y = \frac{1}{3}y - 8
$$

Step 2: Subtract $ \frac{1}{3}y $ from both sides:
$$
25 = -8
$$

✘ This is a contradiction → No solution

Wait — this can't be. Let's double-check:

Right side: $ -\frac{1}{3}y - 8 + \frac{2}{3}y = (-\frac{1}{3} + \frac{2}{3})y - 8 = \frac{1}{3}y - 8 $

So:
$$
25 + \frac{1}{3}y = \frac{1}{3}y - 8
$$

Subtract $ \frac{1}{3}y $:
$$
25 = -8
$$

This is false, so no value of $ y $ satisfies it.

✘ No solution

But let’s check if I misread the problem.

Original: $ 25 + \frac{1}{3}y = -\frac{1}{3}y - 8 + \frac{2}{3}y $

Yes, that’s correct.

So final answer: No solution

---

5) $ -3(x - 7) = -15 - 4x + 7x $

Step 1: Distribute on the left:
$$
-3x + 21 = -15 + 3x
$$

Step 2: Subtract $ 3x $ from both sides:
$$
-6x + 21 = -15
$$

Step 3: Subtract 21:
$$
-6x = -36
$$

Step 4: Divide by -6:
$$
x = 6
$$

✔ Check:
Left: $ -3(6 - 7) = -3(-1) = 3 $
Right: $ -15 - 4(6) + 7(6) = -15 - 24 + 42 = 3 $
✔️ Correct

---

6) $ -2(c - 5) = 10 - 2c $

Step 1: Distribute:
$$
-2c + 10 = 10 - 2c
$$

Step 2: Add $ 2c $ to both sides:
$$
10 = 10
$$

✔ This is always true → Infinite solutions (all real numbers)

Because both sides are identical after simplification.

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7) $ -9(h - 1) = 5 - (h - 8) $

Step 1: Distribute:
Left: $ -9h + 9 $
Right: $ 5 - h + 8 = -h + 13 $

So:
$$
-9h + 9 = -h + 13
$$

Step 2: Add $ 9h $ to both sides:
$$
9 = 8h + 13
$$

Step 3: Subtract 13:
$$
-4 = 8h
$$

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

✔ Check:
Left: $ -9(-\frac{1}{2} - 1) = -9(-\frac{3}{2}) = \frac{27}{2} = 13.5 $
Right: $ 5 - (-\frac{1}{2} - 8) = 5 - (-8.5) = 5 + 8.5 = 13.5 $
✔️ Correct

---

8) $ -k + 2(k - 6.2) = -10k - 4.7 $

Step 1: Distribute:
$$
-k + 2k - 12.4 = -10k - 4.7
\Rightarrow k - 12.4 = -10k - 4.7
$$

Step 2: Add $ 10k $ to both sides:
$$
11k - 12.4 = -4.7
$$

Step 3: Add 12.4:
$$
11k = 7.7
$$

Step 4: Divide by 11:
$$
k = \frac{7.7}{11} = 0.7
$$

✔ Check:
Left: $ -0.7 + 2(0.7 - 6.2) = -0.7 + 2(-5.5) = -0.7 - 11 = -11.7 $
Right: $ -10(0.7) - 4.7 = -7 - 4.7 = -11.7 $
✔️ Correct

---

9) $ 4.1(h - 5) = .1 - (h - 10) $

Step 1: Distribute:
Left: $ 4.1h - 20.5 $
Right: $ 0.1 - h + 10 = -h + 10.1 $

So:
$$
4.1h - 20.5 = -h + 10.1
$$

Step 2: Add $ h $ to both sides:
$$
5.1h - 20.5 = 10.1
$$

Step 3: Add 20.5:
$$
5.1h = 30.6
$$

Step 4: Divide by 5.1:
$$
h = \frac{30.6}{5.1} = 6
$$

✔ Check:
Left: $ 4.1(6 - 5) = 4.1(1) = 4.1 $
Right: $ 0.1 - (6 - 10) = 0.1 - (-4) = 0.1 + 4 = 4.1 $
✔️ Correct

---

10) $ \frac{1}{2}k + 2(k - 5) = -5(k - 1) $

Step 1: Distribute:
Left: $ \frac{1}{2}k + 2k - 10 = \frac{5}{2}k - 10 $
Right: $ -5k + 5 $

So:
$$
\frac{5}{2}k - 10 = -5k + 5
$$

Step 2: Add $ 5k $ to both sides:
$$
\frac{5}{2}k + 5k - 10 = 5
\Rightarrow \left(\frac{5}{2} + \frac{10}{2}\right)k - 10 = 5
\Rightarrow \frac{15}{2}k - 10 = 5
$$

Step 3: Add 10:
$$
\frac{15}{2}k = 15
$$

Step 4: Multiply both sides by $ \frac{2}{15} $:
$$
k = 15 \cdot \frac{2}{15} = 2
$$

✔ Check:
Left: $ \frac{1}{2}(2) + 2(2 - 5) = 1 + 2(-3) = 1 - 6 = -5 $
Right: $ -5(2 - 1) = -5(1) = -5 $
✔️ Correct

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✔ Final Answers:

| Problem | Answer |
|--------|--------|
| 1) | $ m = -1 $ |
| 2) | $ x = \frac{7}{8} $ |
| 3) | $ y = 16 $ |
| 4) | No solution |
| 5) | $ x = 6 $ |
| 6) | All real numbers (infinite solutions) |
| 7) | $ h = -\frac{1}{2} $ |
| 8) | $ k = 0.7 $ |
| 9) | $ h = 6 $ |
| 10) | $ k = 2 $ |

All checked and verified! 🎉