Linear Equation in One Variable Worksheet - 2: Practice problems for solving equations with one variable.

Let's solve each of these linear equations in one variable step by step. I'll go through all 29 problems and provide the solution with clear explanations.

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(1) $ 8x = 6x + 10 $

Subtract $6x$ from both sides:
$$
8x - 6x = 10 \Rightarrow 2x = 10
$$
Divide by 2:
$$
x = 5
$$

✔ Answer: $ x = 5 $

---

(2) $ 4 = 5x - 6 $

Add 6 to both sides:
$$
4 + 6 = 5x \Rightarrow 10 = 5x
$$
Divide by 5:
$$
x = 2
$$

✔ Answer: $ x = 2 $

---

(3) $ 13y = -12y + 100 $

Add $12y$ to both sides:
$$
13y + 12y = 100 \Rightarrow 25y = 100
$$
Divide by 25:
$$
y = 4
$$

✔ Answer: $ y = 4 $

---

(4) $ 18x = -13x + 62 $

Add $13x$ to both sides:
$$
18x + 13x = 62 \Rightarrow 31x = 62
$$
Divide by 31:
$$
x = 2
$$

✔ Answer: $ x = 2 $

---

(5) $ 5x + (-3) = 12 $

Add 3 to both sides:
$$
5x = 15
$$
Divide by 5:
$$
x = 3
$$

✔ Answer: $ x = 3 $

---

(6) $ 3(x + 1) = 6 $

Distribute:
$$
3x + 3 = 6
$$
Subtract 3:
$$
3x = 3 \Rightarrow x = 1
$$

✔ Answer: $ x = 1 $

---

(7) $ 7(m - 9) = 35 $

Divide both sides by 7:
$$
m - 9 = 5
$$
Add 9:
$$
m = 14
$$

✔ Answer: $ m = 14 $

---

(8) $ 8(x + 3) + 2 = 42 $

Distribute:
$$
8x + 24 + 2 = 42 \Rightarrow 8x + 26 = 42
$$
Subtract 26:
$$
8x = 16 \Rightarrow x = 2
$$

✔ Answer: $ x = 2 $

---

(9) $ 16 - 3(x - 7) = -14 $

Distribute:
$$
16 - 3x + 21 = -14 \Rightarrow -3x + 37 = -14
$$
Subtract 37:
$$
-3x = -51 \Rightarrow x = 17
$$

✔ Answer: $ x = 17 $

---

(10) $ 3(x + 5) = 15 $

Divide both sides by 3:
$$
x + 5 = 5 \Rightarrow x = 0
$$

✔ Answer: $ x = 0 $

---

(11) $ 12(3 - x) = 48 $

Divide both sides by 12:
$$
3 - x = 4
$$
Subtract 3:
$$
-x = 1 \Rightarrow x = -1
$$

✔ Answer: $ x = -1 $

---

(12) $ 5x + 8(2x - 9) = 54 $

Distribute:
$$
5x + 16x - 72 = 54 \Rightarrow 21x - 72 = 54
$$
Add 72:
$$
21x = 126 \Rightarrow x = 6
$$

✔ Answer: $ x = 6 $

---

(13) $ \frac{x}{6} = 5 $

Multiply both sides by 6:
$$
x = 30
$$

✔ Answer: $ x = 30 $

---

(14) $ \frac{m}{3} = 4 $

Multiply both sides by 3:
$$
m = 12
$$

✔ Answer: $ m = 12 $

---

(15) $ \frac{m}{4} = \frac{1}{2} $

Multiply both sides by 4:
$$
m = 2
$$

✔ Answer: $ m = 2 $

---

(16) $ \frac{x}{11} = 6 $

Multiply both sides by 11:
$$
x = 66
$$

✔ Answer: $ x = 66 $

---

(17) $ \frac{x}{5} = 7 $

Multiply both sides by 5:
$$
x = 35
$$

✔ Answer: $ x = 35 $

---

(18) $ \frac{x}{7} = 4.5 $

Multiply both sides by 7:
$$
x = 4.5 \times 7 = 31.5
$$

✔ Answer: $ x = 31.5 $ or $ \frac{63}{2} $

---

(19) $ \frac{r}{9} = -11 $

Multiply both sides by 9:
$$
r = -99
$$

✔ Answer: $ r = -99 $

---

(20) $ \frac{x}{-4} = \frac{1}{8} $

Multiply both sides by $-4$:
$$
x = -4 \times \frac{1}{8} = -\frac{1}{2}
$$

✔ Answer: $ x = -\frac{1}{2} $

---

(21) $ \frac{x}{-4} = \frac{3}{4} $

Multiply both sides by $-4$:
$$
x = -4 \times \frac{3}{4} = -3
$$

✔ Answer: $ x = -3 $

---

(22) $ \frac{x}{2} = -39 $

Multiply both sides by 2:
$$
x = -78
$$

✔ Answer: $ x = -78 $

---

(23) $ \frac{5z - 7}{3z} = 2 $

Multiply both sides by $3z$:
$$
5z - 7 = 6z
$$
Subtract $5z$:
$$
-7 = z \Rightarrow z = -7
$$

✔ Answer: $ z = -7 $

---

(24) $ \frac{2y + 5}{y + 4} = 1 $

Multiply both sides by $y + 4$:
$$
2y + 5 = y + 4
$$
Subtract $y$:
$$
y + 5 = 4 \Rightarrow y = -1
$$

✔ Answer: $ y = -1 $

---

(25) $ \frac{2x + 1}{3x - 2} = \frac{5}{9} $

Cross-multiply:
$$
9(2x + 1) = 5(3x - 2)
$$
Expand:
$$
18x + 9 = 15x - 10
$$
Subtract $15x$:
$$
3x + 9 = -10 \Rightarrow 3x = -19 \Rightarrow x = -\frac{19}{3}
$$

✔ Answer: $ x = -\frac{19}{3} $

---

(26) $ \frac{2x}{3x + 1} = -3 $

Multiply both sides by $3x + 1$:
$$
2x = -3(3x + 1) = -9x - 3
$$
Add $9x$:
$$
11x = -3 \Rightarrow x = -\frac{3}{11}
$$

✔ Answer: $ x = -\frac{3}{11} $

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(27) $ \frac{5y}{3} + \frac{2}{5} = 1 $

Subtract $\frac{2}{5}$:
$$
\frac{5y}{3} = 1 - \frac{2}{5} = \frac{3}{5}
$$
Multiply both sides by 3:
$$
5y = \frac{9}{5}
$$
Divide by 5:
$$
y = \frac{9}{25}
$$

✔ Answer: $ y = \frac{9}{25} $

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(28) $ \frac{x}{2} + \frac{x}{3} + \frac{x}{4} = 13 $

Find LCM of denominators: LCM(2,3,4) = 12

Convert:
$$
\frac{6x}{12} + \frac{4x}{12} + \frac{3x}{12} = 13 \Rightarrow \frac{13x}{12} = 13
$$
Multiply both sides by 12:
$$
13x = 156 \Rightarrow x = 12
$$

✔ Answer: $ x = 12 $

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(29) $ \frac{2x}{3} - \frac{3x}{8} = \frac{7}{12} $

LCM of 3, 8, 12 is 24

Convert:
$$
\frac{16x}{24} - \frac{9x}{24} = \frac{7}{12} \Rightarrow \frac{7x}{24} = \frac{7}{12}
$$
Multiply both sides by 24:
$$
7x = 14 \Rightarrow x = 2
$$

✔ Answer: $ x = 2 $

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✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| (1) | $ x = 5 $ |
| (2) | $ x = 2 $ |
| (3) | $ y = 4 $ |
| (4) | $ x = 2 $ |
| (5) | $ x = 3 $ |
| (6) | $ x = 1 $ |
| (7) | $ m = 14 $ |
| (8) | $ x = 2 $ |
| (9) | $ x = 17 $ |
| (10) | $ x = 0 $ |
| (11) | $ x = -1 $ |
| (12) | $ x = 6 $ |
| (13) | $ x = 30 $ |
| (14) | $ m = 12 $ |
| (15) | $ m = 2 $ |
| (16) | $ x = 66 $ |
| (17) | $ x = 35 $ |
| (18) | $ x = 31.5 $ or $ \frac{63}{2} $ |
| (19) | $ r = -99 $ |
| (20) | $ x = -\frac{1}{2} $ |
| (21) | $ x = -3 $ |
| (22) | $ x = -78 $ |
| (23) | $ z = -7 $ |
| (24) | $ y = -1 $ |
| (25) | $ x = -\frac{19}{3} $ |
| (26) | $ x = -\frac{3}{11} $ |
| (27) | $ y = \frac{9}{25} $ |
| (28) | $ x = 12 $ |
| (29) | $ x = 2 $ |

Let me know if you'd like a printable version or explanation for any specific problem!