Pre-Algebra equations worksheet for solving linear equations with variables.

Problem Overview:

The image contains a set of 18 algebraic equations, each requiring the solution for a variable. The task is to solve each equation step-by-step and provide the value of the variable.

Solution Approach:

We will solve each equation systematically by isolating the variable on one side of the equation. Here are the steps for solving linear equations:
1. Simplify both sides of the equation.
2. Use inverse operations (addition/subtraction, multiplication/division) to isolate the variable.
3. Solve for the variable.

Let's solve each equation one by one.

---

Equation ①: \( 4a + 15 = 31 \)

1. Subtract 15 from both sides:
\[
4a + 15 - 15 = 31 - 15
\]
\[
4a = 16
\]
2. Divide both sides by 4:
\[
\frac{4a}{4} = \frac{16}{4}
\]
\[
a = 4
\]

Answer: \( a = 4 \)

---

Equation ②: \( 5x + 8 = 18 \)

1. Subtract 8 from both sides:
\[
5x + 8 - 8 = 18 - 8
\]
\[
5x = 10
\]
2. Divide both sides by 5:
\[
\frac{5x}{5} = \frac{10}{5}
\]
\[
x = 2
\]

Answer: \( x = 2 \)

---

Equation ③: \( 3a + 12 = 24 \)

1. Subtract 12 from both sides:
\[
3a + 12 - 12 = 24 - 12
\]
\[
3a = 12
\]
2. Divide both sides by 3:
\[
\frac{3a}{3} = \frac{12}{3}
\]
\[
a = 4
\]

Answer: \( a = 4 \)

---

Equation ④: \( 5c + 2 = 22 \)

1. Subtract 2 from both sides:
\[
5c + 2 - 2 = 22 - 2
\]
\[
5c = 20
\]
2. Divide both sides by 5:
\[
\frac{5c}{5} = \frac{20}{5}
\]
\[
c = 4
\]

Answer: \( c = 4 \)

---

Equation ⑤: \( 4x - 9 = 7 \)

1. Add 9 to both sides:
\[
4x - 9 + 9 = 7 + 9
\]
\[
4x = 16
\]
2. Divide both sides by 4:
\[
\frac{4x}{4} = \frac{16}{4}
\]
\[
x = 4
\]

Answer: \( x = 4 \)

---

Equation ⑥: \( x + 1 = 4 \)

1. Subtract 1 from both sides:
\[
x + 1 - 1 = 4 - 1
\]
\[
x = 3
\]

Answer: \( x = 3 \)

---

Equation ⑦: \( 6x - 3 = 3 \)

1. Add 3 to both sides:
\[
6x - 3 + 3 = 3 + 3
\]
\[
6x = 6
\]
2. Divide both sides by 6:
\[
\frac{6x}{6} = \frac{6}{6}
\]
\[
x = 1
\]

Answer: \( x = 1 \)

---

Equation ⑧: \( 2a + 2 = 8 \)

1. Subtract 2 from both sides:
\[
2a + 2 - 2 = 8 - 2
\]
\[
2a = 6
\]
2. Divide both sides by 2:
\[
\frac{2a}{2} = \frac{6}{2}
\]
\[
a = 3
\]

Answer: \( a = 3 \)

---

Equation ⑨: \( 6a + 8 = 32 \)

1. Subtract 8 from both sides:
\[
6a + 8 - 8 = 32 - 8
\]
\[
6a = 24
\]
2. Divide both sides by 6:
\[
\frac{6a}{6} = \frac{24}{6}
\]
\[
a = 4
\]

Answer: \( a = 4 \)

---

Equation ⑩: \( 4x - 6 = 2 \)

1. Add 6 to both sides:
\[
4x - 6 + 6 = 2 + 6
\]
\[
4x = 8
\]
2. Divide both sides by 4:
\[
\frac{4x}{4} = \frac{8}{4}
\]
\[
x = 2
\]

Answer: \( x = 2 \)

---

Equation ⑪: \( 3x - 2 = 10 \)

1. Add 2 to both sides:
\[
3x - 2 + 2 = 10 + 2
\]
\[
3x = 12
\]
2. Divide both sides by 3:
\[
\frac{3x}{3} = \frac{12}{3}
\]
\[
x = 4
\]

Answer: \( x = 4 \)

---

Equation ⑫: \( a + 1 = 5 \)

1. Subtract 1 from both sides:
\[
a + 1 - 1 = 5 - 1
\]
\[
a = 4
\]

Answer: \( a = 4 \)

---

Equation ⑬: \( c + 1 = 3 \)

1. Subtract 1 from both sides:
\[
c + 1 - 1 = 3 - 1
\]
\[
c = 2
\]

Answer: \( c = 2 \)

---

Equation ⑭: \( 3b - 4 = 14 \)

1. Add 4 to both sides:
\[
3b - 4 + 4 = 14 + 4
\]
\[
3b = 18
\]
2. Divide both sides by 3:
\[
\frac{3b}{3} = \frac{18}{3}
\]
\[
b = 6
\]

Answer: \( b = 6 \)

---

Equation ⑮: \( 6a - 13 = 11 \)

1. Add 13 to both sides:
\[
6a - 13 + 13 = 11 + 13
\]
\[
6a = 24
\]
2. Divide both sides by 6:
\[
\frac{6a}{6} = \frac{24}{6}
\]
\[
a = 4
\]

Answer: \( a = 4 \)

---

Equation ⑯: \( 6b - 36 = 0 \)

1. Add 36 to both sides:
\[
6b - 36 + 36 = 0 + 36
\]
\[
6b = 36
\]
2. Divide both sides by 6:
\[
\frac{6b}{6} = \frac{36}{6}
\]
\[
b = 6
\]

Answer: \( b = 6 \)

---

Equation ⑰: \( 4b - 9 = 15 \)

1. Add 9 to both sides:
\[
4b - 9 + 9 = 15 + 9
\]
\[
4b = 24
\]
2. Divide both sides by 4:
\[
\frac{4b}{4} = \frac{24}{4}
\]
\[
b = 6
\]

Answer: \( b = 6 \)

---

Equation ⑱: \( 6b - 12 = 18 \)

1. Add 12 to both sides:
\[
6b - 12 + 12 = 18 + 12
\]
\[
6b = 30
\]
2. Divide both sides by 6:
\[
\frac{6b}{6} = \frac{30}{6}
\]
\[
b = 5
\]

Answer: \( b = 5 \)

---

Final Answers:

\[
\boxed{
\begin{array}{ll}
① & a = 4 \\
② & x = 2 \\
③ & a = 4 \\
④ & c = 4 \\
⑤ & x = 4 \\
⑥ & x = 3 \\
⑦ & x = 1 \\
⑧ & a = 3 \\
⑨ & a = 4 \\
⑩ & x = 2 \\
⑪ & x = 4 \\
⑫ & a = 4 \\
⑬ & c = 2 \\
⑭ & b = 6 \\
⑮ & a = 4 \\
⑯ & b = 6 \\
⑰ & b = 6 \\
⑱ & b = 5 \\
\end{array}
}
\]