Practice solving for unknowns with these 20 short linear equation problems designed for algebra students.

Problem: Solve for the unknown in the following equations.

The worksheet contains 20 linear equations in one variable. Below, I will solve each equation step by step.

---

#### Equation (1): \( 8x = 6x + 10 \)
1. Subtract \( 6x \) from both sides:
\[
8x - 6x = 10
\]
2. Simplify:
\[
2x = 10
\]
3. Divide both sides by 2:
\[
x = 5
\]

Answer: \( x = 5 \)

---

#### Equation (2): \( 4 = 5x - 6 \)
1. Add 6 to both sides:
\[
4 + 6 = 5x
\]
2. Simplify:
\[
10 = 5x
\]
3. Divide both sides by 5:
\[
x = 2
\]

Answer: \( x = 2 \)

---

#### Equation (3): \( 13y = -12y + 100 \)
1. Add \( 12y \) to both sides:
\[
13y + 12y = 100
\]
2. Simplify:
\[
25y = 100
\]
3. Divide both sides by 25:
\[
y = 4
\]

Answer: \( y = 4 \)

---

#### Equation (4): \( 18x = -13x + 62 \)
1. Add \( 13x \) to both sides:
\[
18x + 13x = 62
\]
2. Simplify:
\[
31x = 62
\]
3. Divide both sides by 31:
\[
x = 2
\]

Answer: \( x = 2 \)

---

#### Equation (5): \( 5x - 3 = 12 \)
1. Add 3 to both sides:
\[
5x = 15
\]
2. Divide both sides by 5:
\[
x = 3
\]

Answer: \( x = 3 \)

---

#### Equation (6): \( 3(x + 1) = 6 \)
1. Distribute the 3:
\[
3x + 3 = 6
\]
2. Subtract 3 from both sides:
\[
3x = 3
\]
3. Divide both sides by 3:
\[
x = 1
\]

Answer: \( x = 1 \)

---

#### Equation (7): \( 7(m - 9) = 35 \)
1. Distribute the 7:
\[
7m - 63 = 35
\]
2. Add 63 to both sides:
\[
7m = 98
\]
3. Divide both sides by 7:
\[
m = 14
\]

Answer: \( m = 14 \)

---

#### Equation (8): \( 8(x + 3) + 2 = 42 \)
1. Distribute the 8:
\[
8x + 24 + 2 = 42
\]
2. Simplify:
\[
8x + 26 = 42
\]
3. Subtract 26 from both sides:
\[
8x = 16
\]
4. Divide both sides by 8:
\[
x = 2
\]

Answer: \( x = 2 \)

---

#### Equation (9): \( 16 - 3(x - 7) = -14 \)
1. Distribute the \(-3\):
\[
16 - 3x + 21 = -14
\]
2. Simplify:
\[
37 - 3x = -14
\]
3. Subtract 37 from both sides:
\[
-3x = -51
\]
4. Divide both sides by \(-3\):
\[
x = 17
\]

Answer: \( x = 17 \)

---

#### Equation (10): \( 3(x + 5) = 15 \)
1. Distribute the 3:
\[
3x + 15 = 15
\]
2. Subtract 15 from both sides:
\[
3x = 0
\]
3. Divide both sides by 3:
\[
x = 0
\]

Answer: \( x = 0 \)

---

#### Equation (11): \( 12(3 - x) = 48 \)
1. Distribute the 12:
\[
36 - 12x = 48
\]
2. Subtract 36 from both sides:
\[
-12x = 12
\]
3. Divide both sides by \(-12\):
\[
x = -1
\]

Answer: \( x = -1 \)

---

#### Equation (12): \( 5x + 8(2x - 9) = 54 \)
1. Distribute the 8:
\[
5x + 16x - 72 = 54
\]
2. Combine like terms:
\[
21x - 72 = 54
\]
3. Add 72 to both sides:
\[
21x = 126
\]
4. Divide both sides by 21:
\[
x = 6
\]

Answer: \( x = 6 \)

---

#### Equation (13): \( \frac{x}{6} = 5 \)
1. Multiply both sides by 6:
\[
x = 30
\]

Answer: \( x = 30 \)

---

#### Equation (14): \( \frac{m}{3} = 4 \)
1. Multiply both sides by 3:
\[
m = 12
\]

Answer: \( m = 12 \)

---

#### Equation (15): \( \frac{m}{4} = \frac{1}{2} \)
1. Multiply both sides by 4:
\[
m = 2
\]

Answer: \( m = 2 \)

---

#### Equation (16): \( \frac{x}{11} = 6 \)
1. Multiply both sides by 11:
\[
x = 66
\]

Answer: \( x = 66 \)

---

#### Equation (17): \( \frac{x}{5} = 7 \)
1. Multiply both sides by 5:
\[
x = 35
\]

Answer: \( x = 35 \)

---

#### Equation (18): \( \frac{x}{7} = 4.5 \)
1. Multiply both sides by 7:
\[
x = 31.5
\]

Answer: \( x = 31.5 \)

---

#### Equation (19): \( \frac{r}{9} = -11 \)
1. Multiply both sides by 9:
\[
r = -99
\]

Answer: \( r = -99 \)

---

#### Equation (20): \( \frac{x}{-4} = \frac{1}{8} \)
1. Multiply both sides by \(-4\):
\[
x = \frac{-4}{8}
\]
2. Simplify:
\[
x = -\frac{1}{2}
\]

Answer: \( x = -\frac{1}{2} \)

---

Final Answers:

\[
\boxed{
\begin{aligned}
&1.~ x = 5 \\
&2.~ x = 2 \\
&3.~ y = 4 \\
&4.~ x = 2 \\
&5.~ x = 3 \\
&6.~ x = 1 \\
&7.~ m = 14 \\
&8.~ x = 2 \\
&9.~ x = 17 \\
&10.~ x = 0 \\
&11.~ x = -1 \\
&12.~ x = 6 \\
&13.~ x = 30 \\
&14.~ m = 12 \\
&15.~ m = 2 \\
&16.~ x = 66 \\
&17.~ x = 35 \\
&18.~ x = 31.5 \\
&19.~ r = -99 \\
&20.~ x = -\frac{1}{2}
\end{aligned}
}
\]