Practice solving for variables with this comprehensive linear equations worksheet designed for middle school math students.

Linear Equations Worksheet Solution

The task is to solve each of the given linear equations. Below are the solutions with step-by-step explanations.

---

#### 1 a. \( 8 = \frac{z}{6} \)

1. To isolate \( z \), multiply both sides of the equation by 6:
\[
8 \times 6 = \frac{z}{6} \times 6
\]
2. Simplify:
\[
48 = z
\]

Solution: \( z = 48 \)

---

#### 1 b. \( c + 1 = 1 \)

1. To isolate \( c \), subtract 1 from both sides:
\[
c + 1 - 1 = 1 - 1
\]
2. Simplify:
\[
c = 0
\]

Solution: \( c = 0 \)

---

#### 2 a. \( 11 = \frac{x}{8} \)

1. To isolate \( x \), multiply both sides of the equation by 8:
\[
11 \times 8 = \frac{x}{8} \times 8
\]
2. Simplify:
\[
88 = x
\]

Solution: \( x = 88 \)

---

#### 2 b. \( 2 + y = 11 \)

1. To isolate \( y \), subtract 2 from both sides:
\[
2 + y - 2 = 11 - 2
\]
2. Simplify:
\[
y = 9
\]

Solution: \( y = 9 \)

---

#### 3 a. \( b - 11 = 11 \)

1. To isolate \( b \), add 11 to both sides:
\[
b - 11 + 11 = 11 + 11
\]
2. Simplify:
\[
b = 22
\]

Solution: \( b = 22 \)

---

#### 3 b. \( c + 5 = 5 \)

1. To isolate \( c \), subtract 5 from both sides:
\[
c + 5 - 5 = 5 - 5
\]
2. Simplify:
\[
c = 0
\]

Solution: \( c = 0 \)

---

#### 4 a. \( 5z = 4 \)

1. To isolate \( z \), divide both sides by 5:
\[
\frac{5z}{5} = \frac{4}{5}
\]
2. Simplify:
\[
z = \frac{4}{5}
\]

Solution: \( z = \frac{4}{5} \)

---

#### 4 b. \( 5 - b = 1 \)

1. To isolate \( b \), subtract 5 from both sides:
\[
5 - b - 5 = 1 - 5
\]
2. Simplify:
\[
-b = -4
\]
3. Multiply both sides by -1 to solve for \( b \):
\[
b = 4
\]

Solution: \( b = 4 \)

---

#### 5 a. \( 5 = 9s \)

1. To isolate \( s \), divide both sides by 9:
\[
\frac{5}{9} = \frac{9s}{9}
\]
2. Simplify:
\[
s = \frac{5}{9}
\]

Solution: \( s = \frac{5}{9} \)

---

#### 5 b. \( 3 = 7p \)

1. To isolate \( p \), divide both sides by 7:
\[
\frac{3}{7} = \frac{7p}{7}
\]
2. Simplify:
\[
p = \frac{3}{7}
\]

Solution: \( p = \frac{3}{7} \)

---

#### 6 a. \( 11 = p + 1 \)

1. To isolate \( p \), subtract 1 from both sides:
\[
11 - 1 = p + 1 - 1
\]
2. Simplify:
\[
p = 10
\]

Solution: \( p = 10 \)

---

#### 6 b. \( \frac{c}{11} = 2 \)

1. To isolate \( c \), multiply both sides by 11:
\[
\frac{c}{11} \times 11 = 2 \times 11
\]
2. Simplify:
\[
c = 22
\]

Solution: \( c = 22 \)

---

Final Answers:

\[
\boxed{
\begin{array}{ll}
1 \text{ a. } z = 48 & 1 \text{ b. } c = 0 \\
2 \text{ a. } x = 88 & 2 \text{ b. } y = 9 \\
3 \text{ a. } b = 22 & 3 \text{ b. } c = 0 \\
4 \text{ a. } z = \frac{4}{5} & 4 \text{ b. } b = 4 \\
5 \text{ a. } s = \frac{5}{9} & 5 \text{ b. } p = \frac{3}{7} \\
6 \text{ a. } p = 10 & 6 \text{ b. } c = 22 \\
\end{array}
}
\]