Solving Linear Equations (Including Negative Values) -- Form a/x ... - Free Printable

Problem: Solve each of the given simple linear equations for the variable.

The equations are as follows:

1. \( \frac{50}{u} - (-2) = 7 \)
2. \( -1 - \frac{-18}{c} = 8 \)
3. \( \frac{-12}{b} - 10 = -6 \)
4. \( 2 - \frac{45}{y} = -7 \)
5. \( \frac{70}{u} + 1 = 8 \)
6. \( 4 + \frac{15}{v} = 7 \)
7. \( \frac{-21}{u} + 8 = 11 \)
8. \( \frac{-24}{b} - (-5) = 9 \)
9. \( \frac{-10}{a} + 4 = 9 \)
10. \( \frac{20}{v} - 9 = -5 \)
11. \( 9 + \frac{30}{d} = 6 \)
12. \( \frac{21}{z} + 3 = 6 \)
13. \( \frac{35}{y} - 9 = -14 \)
14. \( \frac{-32}{z} + 6 = 14 \)
15. \( 3 - \frac{-32}{c} = -5 \)

Solution:

#### Equation 1: \( \frac{50}{u} - (-2) = 7 \)

1. Simplify the equation:
\[
\frac{50}{u} + 2 = 7
\]
2. Subtract 2 from both sides:
\[
\frac{50}{u} = 5
\]
3. Multiply both sides by \( u \):
\[
50 = 5u
\]
4. Divide both sides by 5:
\[
u = 10
\]

Answer: \( u = 10 \)

---

#### Equation 2: \( -1 - \frac{-18}{c} = 8 \)

1. Simplify the equation:
\[
-1 + \frac{18}{c} = 8
\]
2. Add 1 to both sides:
\[
\frac{18}{c} = 9
\]
3. Multiply both sides by \( c \):
\[
18 = 9c
\]
4. Divide both sides by 9:
\[
c = 2
\]

Answer: \( c = 2 \)

---

#### Equation 3: \( \frac{-12}{b} - 10 = -6 \)

1. Add 10 to both sides:
\[
\frac{-12}{b} = 4
\]
2. Multiply both sides by \( b \):
\[
-12 = 4b
\]
3. Divide both sides by 4:
\[
b = -3
\]

Answer: \( b = -3 \)

---

#### Equation 4: \( 2 - \frac{45}{y} = -7 \)

1. Subtract 2 from both sides:
\[
-\frac{45}{y} = -9
\]
2. Multiply both sides by \( y \):
\[
-45 = -9y
\]
3. Divide both sides by -9:
\[
y = 5
\]

Answer: \( y = 5 \)

---

#### Equation 5: \( \frac{70}{u} + 1 = 8 \)

1. Subtract 1 from both sides:
\[
\frac{70}{u} = 7
\]
2. Multiply both sides by \( u \):
\[
70 = 7u
\]
3. Divide both sides by 7:
\[
u = 10
\]

Answer: \( u = 10 \)

---

#### Equation 6: \( 4 + \frac{15}{v} = 7 \)

1. Subtract 4 from both sides:
\[
\frac{15}{v} = 3
\]
2. Multiply both sides by \( v \):
\[
15 = 3v
\]
3. Divide both sides by 3:
\[
v = 5
\]

Answer: \( v = 5 \)

---

#### Equation 7: \( \frac{-21}{u} + 8 = 11 \)

1. Subtract 8 from both sides:
\[
\frac{-21}{u} = 3
\]
2. Multiply both sides by \( u \):
\[
-21 = 3u
\]
3. Divide both sides by 3:
\[
u = -7
\]

Answer: \( u = -7 \)

---

#### Equation 8: \( \frac{-24}{b} - (-5) = 9 \)

1. Simplify the equation:
\[
\frac{-24}{b} + 5 = 9
\]
2. Subtract 5 from both sides:
\[
\frac{-24}{b} = 4
\]
3. Multiply both sides by \( b \):
\[
-24 = 4b
\]
4. Divide both sides by 4:
\[
b = -6
\]

Answer: \( b = -6 \)

---

#### Equation 9: \( \frac{-10}{a} + 4 = 9 \)

1. Subtract 4 from both sides:
\[
\frac{-10}{a} = 5
\]
2. Multiply both sides by \( a \):
\[
-10 = 5a
\]
3. Divide both sides by 5:
\[
a = -2
\]

Answer: \( a = -2 \)

---

#### Equation 10: \( \frac{20}{v} - 9 = -5 \)

1. Add 9 to both sides:
\[
\frac{20}{v} = 4
\]
2. Multiply both sides by \( v \):
\[
20 = 4v
\]
3. Divide both sides by 4:
\[
v = 5
\]

Answer: \( v = 5 \)

---

#### Equation 11: \( 9 + \frac{30}{d} = 6 \)

1. Subtract 9 from both sides:
\[
\frac{30}{d} = -3
\]
2. Multiply both sides by \( d \):
\[
30 = -3d
\]
3. Divide both sides by -3:
\[
d = -10
\]

Answer: \( d = -10 \)

---

#### Equation 12: \( \frac{21}{z} + 3 = 6 \)

1. Subtract 3 from both sides:
\[
\frac{21}{z} = 3
\]
2. Multiply both sides by \( z \):
\[
21 = 3z
\]
3. Divide both sides by 3:
\[
z = 7
\]

Answer: \( z = 7 \)

---

#### Equation 13: \( \frac{35}{y} - 9 = -14 \)

1. Add 9 to both sides:
\[
\frac{35}{y} = -5
\]
2. Multiply both sides by \( y \):
\[
35 = -5y
\]
3. Divide both sides by -5:
\[
y = -7
\]

Answer: \( y = -7 \)

---

#### Equation 14: \( \frac{-32}{z} + 6 = 14 \)

1. Subtract 6 from both sides:
\[
\frac{-32}{z} = 8
\]
2. Multiply both sides by \( z \):
\[
-32 = 8z
\]
3. Divide both sides by 8:
\[
z = -4
\]

Answer: \( z = -4 \)

---

#### Equation 15: \( 3 - \frac{-32}{c} = -5 \)

1. Simplify the equation:
\[
3 + \frac{32}{c} = -5
\]
2. Subtract 3 from both sides:
\[
\frac{32}{c} = -8
\]
3. Multiply both sides by \( c \):
\[
32 = -8c
\]
4. Divide both sides by -8:
\[
c = -4
\]

Answer: \( c = -4 \)

---

Final Answers:

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