Solving Linear Equations -- Form a/x ± b = c (A) - Free Printable

Problem: Solve each simple linear equation for the variable.

The task involves solving 15 linear equations, each of which can be solved step-by-step using basic algebraic principles. Below is the solution for each equation:

---

#### Equation 1:
$$
\frac{63}{y} + 1 = 8
$$
1. Subtract 1 from both sides:
$$
\frac{63}{y} = 7
$$
2. Multiply both sides by \( y \):
$$
63 = 7y
$$
3. Divide both sides by 7:
$$
y = 9
$$

Solution: \( y = 9 \)

---

#### Equation 2:
$$
\frac{18}{u} + 5 = 8
$$
1. Subtract 5 from both sides:
$$
\frac{18}{u} = 3
$$
2. Multiply both sides by \( u \):
$$
18 = 3u
$$
3. Divide both sides by 3:
$$
u = 6
$$

Solution: \( u = 6 \)

---

#### Equation 3:
$$
\frac{10}{a} + 7 = 12
$$
1. Subtract 7 from both sides:
$$
\frac{10}{a} = 5
$$
2. Multiply both sides by \( a \):
$$
10 = 5a
$$
3. Divide both sides by 5:
$$
a = 2
$$

Solution: \( a = 2 \)

---

#### Equation 4:
$$
\frac{56}{c} + 6 = 13
$$
1. Subtract 6 from both sides:
$$
\frac{56}{c} = 7
$$
2. Multiply both sides by \( c \):
$$
56 = 7c
$$
3. Divide both sides by 7:
$$
c = 8
$$

Solution: \( c = 8 \)

---

#### Equation 5:
$$
\frac{12}{x} - 4 = 0
$$
1. Add 4 to both sides:
$$
\frac{12}{x} = 4
$$
2. Multiply both sides by \( x \):
$$
12 = 4x
$$
3. Divide both sides by 4:
$$
x = 3
$$

Solution: \( x = 3 \)

---

#### Equation 6:
$$
10 + \frac{36}{z} = 14
$$
1. Subtract 10 from both sides:
$$
\frac{36}{z} = 4
$$
2. Multiply both sides by \( z \):
$$
36 = 4z
$$
3. Divide both sides by 4:
$$
z = 9
$$

Solution: \( z = 9 \)

---

#### Equation 7:
$$
\frac{49}{b} - 5 = 2
$$
1. Add 5 to both sides:
$$
\frac{49}{b} = 7
$$
2. Multiply both sides by \( b \):
$$
49 = 7b
$$
3. Divide both sides by 7:
$$
b = 7
$$

Solution: \( b = 7 \)

---

#### Equation 8:
$$
\frac{4}{y} - 2 = 2
$$
1. Add 2 to both sides:
$$
\frac{4}{y} = 4
$$
2. Multiply both sides by \( y \):
$$
4 = 4y
$$
3. Divide both sides by 4:
$$
y = 1
$$

Solution: \( y = 1 \)

---

#### Equation 9:
$$
7 + \frac{20}{x} = 11
$$
1. Subtract 7 from both sides:
$$
\frac{20}{x} = 4
$$
2. Multiply both sides by \( x \):
$$
20 = 4x
$$
3. Divide both sides by 4:
$$
x = 5
$$

Solution: \( x = 5 \)

---

#### Equation 10:
$$
\frac{6}{c} + 10 = 12
$$
1. Subtract 10 from both sides:
$$
\frac{6}{c} = 2
$$
2. Multiply both sides by \( c \):
$$
6 = 2c
$$
3. Divide both sides by 2:
$$
c = 3
$$

Solution: \( c = 3 \)

---

#### Equation 11:
$$
\frac{40}{z} - 7 = 1
$$
1. Add 7 to both sides:
$$
\frac{40}{z} = 8
$$
2. Multiply both sides by \( z \):
$$
40 = 8z
$$
3. Divide both sides by 8:
$$
z = 5
$$

Solution: \( z = 5 \)

---

#### Equation 12:
$$
8 + \frac{40}{x} = 13
$$
1. Subtract 8 from both sides:
$$
\frac{40}{x} = 5
$$
2. Multiply both sides by \( x \):
$$
40 = 5x
$$
3. Divide both sides by 5:
$$
x = 8
$$

Solution: \( x = 8 \)

---

#### Equation 13:
$$
3 + \frac{18}{b} = 6
$$
1. Subtract 3 from both sides:
$$
\frac{18}{b} = 3
$$
2. Multiply both sides by \( b \):
$$
18 = 3b
$$
3. Divide both sides by 3:
$$
b = 6
$$

Solution: \( b = 6 \)

---

#### Equation 14:
$$
\frac{7}{z} - 3 = 4
$$
1. Add 3 to both sides:
$$
\frac{7}{z} = 7
$$
2. Multiply both sides by \( z \):
$$
7 = 7z
$$
3. Divide both sides by 7:
$$
z = 1
$$

Solution: \( z = 1 \)

---

#### Equation 15:
$$
\frac{20}{b} + 2 = 7
$$
1. Subtract 2 from both sides:
$$
\frac{20}{b} = 5
$$
2. Multiply both sides by \( b \):
$$
20 = 5b
$$
3. Divide both sides by 5:
$$
b = 4
$$

Solution: \( b = 4 \)

---

Final Answers:

$$
\boxed{
\begin{aligned}
1. & \ y = 9 \\
2. & \ u = 6 \\
3. & \ a = 2 \\
4. & \ c = 8 \\
5. & \ x = 3 \\
6. & \ z = 9 \\
7. & \ b = 7 \\
8. & \ y = 1 \\
9. & \ x = 5 \\
10. & \ c = 3 \\
11. & \ z = 5 \\
12. & \ x = 8 \\
13. & \ b = 6 \\
14. & \ z = 1 \\
15. & \ b = 4 \\
\end{aligned}
}
$$