Practice worksheet featuring ten linear equations in one variable for algebra students.

Let's solve each of the linear equations in one variable step by step.

---

1. $ 12 - x = 7 $

Step 1: Subtract 12 from both sides
$$
-x = 7 - 12 \\
-x = -5
$$

Step 2: Multiply both sides by -1
$$
x = 5
$$

✔ Answer: $ x = 5 $

---

2. $ 9 + 6x = 3x + 13 $

Step 1: Subtract $ 3x $ from both sides
$$
9 + 3x = 13
$$

Step 2: Subtract 9 from both sides
$$
3x = 4
$$

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

✔ Answer: $ x = \frac{4}{3} $

---

3. $ 10x + 3 + 10x = 13x - 3 $

Step 1: Combine like terms on the left
$$
20x + 3 = 13x - 3
$$

Step 2: Subtract $ 13x $ from both sides
$$
7x + 3 = -3
$$

Step 3: Subtract 3 from both sides
$$
7x = -6
$$

Step 4: Divide by 7
$$
x = -\frac{6}{7}
$$

✔ Answer: $ x = -\frac{6}{7} $

---

4. $ 0.25(60) + 0.10x = 0.15(60 + x) $

Step 1: Simplify both sides
Left: $ 0.25 \times 60 = 15 $ → $ 15 + 0.10x $
Right: $ 0.15 \times 60 + 0.15x = 9 + 0.15x $

So:
$$
15 + 0.10x = 9 + 0.15x
$$

Step 2: Subtract $ 0.10x $ from both sides
$$
15 = 9 + 0.05x
$$

Step 3: Subtract 9 from both sides
$$
6 = 0.05x
$$

Step 4: Divide by 0.05
$$
x = \frac{6}{0.05} = 120
$$

✔ Answer: $ x = 120 $

---

5. $ 3 = 4(x - 2) + 5 - 4x $

Step 1: Expand the parentheses
$$
3 = 4x - 8 + 5 - 4x
$$

Step 2: Combine like terms
$$
3 = (4x - 4x) + (-8 + 5) = 0x - 3 = -3
$$

So:
$$
3 = -3
$$

✘ This is not true — contradiction.

✔ Answer: No solution (inconsistent equation)

---

6. $ \frac{5y}{9} - 3 = 6 $

Step 1: Add 3 to both sides
$$
\frac{5y}{9} = 9
$$

Step 2: Multiply both sides by 9
$$
5y = 81
$$

Step 3: Divide by 5
$$
y = \frac{81}{5} = 16.2
$$

✔ Answer: $ y = \frac{81}{5} $ or $ 16.2 $

---

7. $ \frac{n}{10} = 9 - \frac{n}{4} $

Step 1: Eliminate denominators by multiplying both sides by the LCM of 10 and 4, which is 20:

$$
20 \cdot \left( \frac{n}{10} \right) = 20 \cdot \left( 9 - \frac{n}{4} \right)
$$

$$
2n = 180 - 5n
$$

Step 2: Add $ 5n $ to both sides
$$
7n = 180
$$

Step 3: Divide by 7
$$
n = \frac{180}{7} \approx 25.714
$$

✔ Answer: $ n = \frac{180}{7} $

---

8. $ 21.1w + 4.6 = 10.9w $

Step 1: Subtract $ 10.9w $ from both sides
$$
10.2w + 4.6 = 0
$$

Step 2: Subtract 4.6 from both sides
$$
10.2w = -4.6
$$

Step 3: Divide by 10.2
$$
w = \frac{-4.6}{10.2} = -\frac{46}{102} = -\frac{23}{51}
$$

✔ Answer: $ w = -\frac{23}{51} $ (or approximately $ -0.451 $)

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9. $ -5(3 - 4x) = -6 + 20x $

Step 1: Expand the left side
$$
-15 + 20x = -6 + 20x
$$

Step 2: Subtract $ 20x $ from both sides
$$
-15 = -6
$$

✘ This is false, so no solution.

✔ Answer: No solution (inconsistent equation)

---

10. $ 9.2z - 4.3 = 50.8 $

Step 1: Add 4.3 to both sides
$$
9.2z = 55.1
$$

Step 2: Divide by 9.2
$$
z = \frac{55.1}{9.2}
$$

To simplify:
$$
z = \frac{551}{92} = 6.0 \text{ (approximately)}
$$

Let’s compute:
$$
9.2 \times 6 = 55.2 \quad \text{(slightly more than 55.1)}
$$
Try:
$$
55.1 ÷ 9.2 ≈ 5.9896...
$$

So:
$$
z = \frac{551}{92} \approx 5.99
$$

But let's keep it exact:

$$
z = \frac{55.1}{9.2} = \frac{551}{92} = \frac{551 \div 1}{92 \div 1} = \frac{551}{92}
$$

We can simplify? Check GCD of 551 and 92.

- 92 = 4×23
- 551 ÷ 23 = 23.95... not divisible
- Try dividing: 551 ÷ 19 = 29 → 19×29 = 551 ✔️
- 92 ÷ 19 = 4.84 → not divisible

So no common factors.

✔ Answer: $ z = \frac{551}{92} $ or approximately $ 5.99 $

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## ✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1 | $ x = 5 $ |
| 2 | $ x = \frac{4}{3} $ |
| 3 | $ x = -\frac{6}{7} $ |
| 4 | $ x = 120 $ |
| 5 | No solution |
| 6 | $ y = \frac{81}{5} $ or $ 16.2 $ |
| 7 | $ n = \frac{180}{7} $ |
| 8 | $ w = -\frac{23}{51} $ |
| 9 | No solution |
| 10 | $ z = \frac{551}{92} $ or $ \approx 5.99 $ |

Let me know if you'd like these as decimals or simplified fractions!