Algebra Worksheet -- Solving Linear Equations (Including Negative ... - Free Printable

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

The task involves solving 15 linear equations. Below, I will solve each equation step by step.

---

#### Equation 1: \( 3b + 9 = -18 \)
1. Subtract 9 from both sides:
\[
3b + 9 - 9 = -18 - 9
\]
\[
3b = -27
\]
2. Divide both sides by 3:
\[
b = \frac{-27}{3}
\]
\[
b = -9
\]

Solution: \( b = -9 \)

---

#### Equation 2: \( 3v + 1 = 22 \)
1. Subtract 1 from both sides:
\[
3v + 1 - 1 = 22 - 1
\]
\[
3v = 21
\]
2. Divide both sides by 3:
\[
v = \frac{21}{3}
\]
\[
v = 7
\]

Solution: \( v = 7 \)

---

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

Solution: \( y = 4 \)

---

#### Equation 4: \( 2z + 1 = 15 \)
1. Subtract 1 from both sides:
\[
2z + 1 - 1 = 15 - 1
\]
\[
2z = 14
\]
2. Divide both sides by 2:
\[
z = \frac{14}{2}
\]
\[
z = 7
\]

Solution: \( z = 7 \)

---

#### Equation 5: \( -2b - (-7) = 11 \)
1. Simplify the left side:
\[
-2b + 7 = 11
\]
2. Subtract 7 from both sides:
\[
-2b + 7 - 7 = 11 - 7
\]
\[
-2b = 4
\]
3. Divide both sides by -2:
\[
b = \frac{4}{-2}
\]
\[
b = -2
\]

Solution: \( b = -2 \)

---

#### Equation 6: \( 3c - 9 = -27 \)
1. Add 9 to both sides:
\[
3c - 9 + 9 = -27 + 9
\]
\[
3c = -18
\]
2. Divide both sides by 3:
\[
c = \frac{-18}{3}
\]
\[
c = -6
\]

Solution: \( c = -6 \)

---

#### Equation 7: \( -3v - 4 = 2 \)
1. Add 4 to both sides:
\[
-3v - 4 + 4 = 2 + 4
\]
\[
-3v = 6
\]
2. Divide both sides by -3:
\[
v = \frac{6}{-3}
\]
\[
v = -2
\]

Solution: \( v = -2 \)

---

#### Equation 8: \( -3c + 8 = -10 \)
1. Subtract 8 from both sides:
\[
-3c + 8 - 8 = -10 - 8
\]
\[
-3c = -18
\]
2. Divide both sides by -3:
\[
c = \frac{-18}{-3}
\]
\[
c = 6
\]

Solution: \( c = 6 \)

---

#### Equation 9: \( 2c - 8 = -18 \)
1. Add 8 to both sides:
\[
2c - 8 + 8 = -18 + 8
\]
\[
2c = -10
\]
2. Divide both sides by 2:
\[
c = \frac{-10}{2}
\]
\[
c = -5
\]

Solution: \( c = -5 \)

---

#### Equation 10: \( 3a - 5 = -23 \)
1. Add 5 to both sides:
\[
3a - 5 + 5 = -23 + 5
\]
\[
3a = -18
\]
2. Divide both sides by 3:
\[
a = \frac{-18}{3}
\]
\[
a = -6
\]

Solution: \( a = -6 \)

---

#### Equation 11: \( -2a - 8 = -4 \)
1. Add 8 to both sides:
\[
-2a - 8 + 8 = -4 + 8
\]
\[
-2a = 4
\]
2. Divide both sides by -2:
\[
a = \frac{4}{-2}
\]
\[
a = -2
\]

Solution: \( a = -2 \)

---

#### Equation 12: \( 3z - 2 = -26 \)
1. Add 2 to both sides:
\[
3z - 2 + 2 = -26 + 2
\]
\[
3z = -24
\]
2. Divide both sides by 3:
\[
z = \frac{-24}{3}
\]
\[
z = -8
\]

Solution: \( z = -8 \)

---

#### Equation 13: \( 2u + 7 = 5 \)
1. Subtract 7 from both sides:
\[
2u + 7 - 7 = 5 - 7
\]
\[
2u = -2
\]
2. Divide both sides by 2:
\[
u = \frac{-2}{2}
\]
\[
u = -1
\]

Solution: \( u = -1 \)

---

#### Equation 14: \( 3a - (-1) = -5 \)
1. Simplify the left side:
\[
3a + 1 = -5
\]
2. Subtract 1 from both sides:
\[
3a + 1 - 1 = -5 - 1
\]
\[
3a = -6
\]
3. Divide both sides by 3:
\[
a = \frac{-6}{3}
\]
\[
a = -2
\]

Solution: \( a = -2 \)

---

#### Equation 15: \( -2u + (-1) = -13 \)
1. Simplify the left side:
\[
-2u - 1 = -13
\]
2. Add 1 to both sides:
\[
-2u - 1 + 1 = -13 + 1
\]
\[
-2u = -12
\]
3. Divide both sides by -2:
\[
u = \frac{-12}{-2}
\]
\[
u = 6
\]

Solution: \( u = 6 \)

---

Final Answers:

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