Multi Step Equations Worksheets - Math Monks - Free Printable

Problem: Solve each multi-step equation involving integers.

We will solve each equation step by step. Let's go through them one by one.

---

#### Equation 1:
\[
-2(3n + 9) = 30
\]

1. Distribute the \(-2\) on the left-hand side:
\[
-2 \cdot 3n + (-2) \cdot 9 = 30
\]
\[
-6n - 18 = 30
\]

2. Add 18 to both sides to isolate the term with \(n\):
\[
-6n - 18 + 18 = 30 + 18
\]
\[
-6n = 48
\]

3. Divide both sides by \(-6\) to solve for \(n\):
\[
n = \frac{48}{-6}
\]
\[
n = -8
\]

Solution for Equation 1:
\[
\boxed{n = -8}
\]

---

#### Equation 2:
\[
9p + 9 - 5 = -12 + 6p
\]

1. Simplify both sides:
\[
9p + 4 = -12 + 6p
\]

2. Subtract \(6p\) from both sides to get all \(p\)-terms on one side:
\[
9p - 6p + 4 = -12
\]
\[
3p + 4 = -12
\]

3. Subtract 4 from both sides to isolate the term with \(p\):
\[
3p + 4 - 4 = -12 - 4
\]
\[
3p = -16
\]

4. Divide both sides by 3 to solve for \(p\):
\[
p = \frac{-16}{3}
\]

Solution for Equation 2:
\[
\boxed{p = -\frac{16}{3}}
\]

---

#### Equation 3:
\[
18n - 6(-4n - 6) = 18 + 8n
\]

1. Distribute the \(-6\) on the left-hand side:
\[
18n - 6 \cdot (-4n) - 6 \cdot (-6) = 18 + 8n
\]
\[
18n + 24n + 36 = 18 + 8n
\]

2. Combine like terms on the left-hand side:
\[
42n + 36 = 18 + 8n
\]

3. Subtract \(8n\) from both sides to get all \(n\)-terms on one side:
\[
42n - 8n + 36 = 18
\]
\[
34n + 36 = 18
\]

4. Subtract 36 from both sides to isolate the term with \(n\):
\[
34n + 36 - 36 = 18 - 36
\]
\[
34n = -18
\]

5. Divide both sides by 34 to solve for \(n\):
\[
n = \frac{-18}{34}
\]
Simplify the fraction:
\[
n = \frac{-9}{17}
\]

Solution for Equation 3:
\[
\boxed{n = -\frac{9}{17}}
\]

---

#### Equation 4:
\[
1 + 8p - 4p + 5 = 2(p - 3) - 2(p - 2)
\]

1. Simplify both sides:
- Left-hand side:
\[
1 + 8p - 4p + 5 = 4p + 6
\]
- Right-hand side:
\[
2(p - 3) - 2(p - 2) = 2p - 6 - (2p - 4) = 2p - 6 - 2p + 4 = -2
\]

So the equation becomes:
\[
4p + 6 = -2
\]

2. Subtract 6 from both sides to isolate the term with \(p\):
\[
4p + 6 - 6 = -2 - 6
\]
\[
4p = -8
\]

3. Divide both sides by 4 to solve for \(p\):
\[
p = \frac{-8}{4}
\]
\[
p = -2
\]

Solution for Equation 4:
\[
\boxed{p = -2}
\]

---

#### Equation 5:
\[
-8x + 16 = -16x + 32
\]

1. Add \(16x\) to both sides to get all \(x\)-terms on one side:
\[
-8x + 16x + 16 = 32
\]
\[
8x + 16 = 32
\]

2. Subtract 16 from both sides to isolate the term with \(x\):
\[
8x + 16 - 16 = 32 - 16
\]
\[
8x = 16
\]

3. Divide both sides by 8 to solve for \(x\):
\[
x = \frac{16}{8}
\]
\[
x = 2
\]

Solution for Equation 5:
\[
\boxed{x = 2}
\]

---

#### Equation 6:
\[
16(5 - 4v) = -5 - v
\]

1. Distribute the 16 on the left-hand side:
\[
16 \cdot 5 - 16 \cdot 4v = -5 - v
\]
\[
80 - 64v = -5 - v
\]

2. Add \(64v\) to both sides to get all \(v\)-terms on one side:
\[
80 - 64v + 64v = -5 - v + 64v
\]
\[
80 = -5 + 63v
\]

3. Add 5 to both sides to isolate the term with \(v\):
\[
80 + 5 = -5 + 63v + 5
\]
\[
85 = 63v
\]

4. Divide both sides by 63 to solve for \(v\):
\[
v = \frac{85}{63}
\]

Solution for Equation 6:
\[
\boxed{v = \frac{85}{63}}
\]

---

#### Equation 7:
\[
-12 = 2(2x + 4) - (7 - 3x)
\]

1. Distribute the 2 and the negative sign:
\[
-12 = 2 \cdot 2x + 2 \cdot 4 - 7 + 3x
\]
\[
-12 = 4x + 8 - 7 + 3x
\]

2. Combine like terms on the right-hand side:
\[
-12 = 4x + 3x + 8 - 7
\]
\[
-12 = 7x + 1
\]

3. Subtract 1 from both sides to isolate the term with \(x\):
\[
-12 - 1 = 7x
\]
\[
-13 = 7x
\]

4. Divide both sides by 7 to solve for \(x\):
\[
x = \frac{-13}{7}
\]

Solution for Equation 7:
\[
\boxed{x = -\frac{13}{7}}
\]

---

#### Equation 8:
\[
-7r - 3r = -13 - 7 - r
\]

1. Combine like terms on both sides:
- Left-hand side:
\[
-7r - 3r = -10r
\]
- Right-hand side:
\[
-13 - 7 = -20
\]
So the equation becomes:
\[
-10r = -20 - r
\]

2. Add \(r\) to both sides to get all \(r\)-terms on one side:
\[
-10r + r = -20
\]
\[
-9r = -20
\]

3. Divide both sides by \(-9\) to solve for \(r\):
\[
r = \frac{-20}{-9}
\]
\[
r = \frac{20}{9}
\]

Solution for Equation 8:
\[
\boxed{r = \frac{20}{9}}
\]

---

#### Equation 9:
\[
16 + 32q + 4 = -32q + 40
\]

1. Combine like terms on the left-hand side:
\[
16 + 4 + 32q = -32q + 40
\]
\[
20 + 32q = -32q + 40
\]

2. Add \(32q\) to both sides to get all \(q\)-terms on one side:
\[
20 + 32q + 32q = 40
\]
\[
20 + 64q = 40
\]

3. Subtract 20 from both sides to isolate the term with \(q\):
\[
20 + 64q - 20 = 40 - 20
\]
\[
64q = 20
\]

4. Divide both sides by 64 to solve for \(q\):
\[
q = \frac{20}{64}
\]
Simplify the fraction:
\[
q = \frac{5}{16}
\]

Solution for Equation 9:
\[
\boxed{q = \frac{5}{16}}
\]

---

#### Equation 10:
\[
-6x + 14 = 1 - 8x
\]

1. Add \(8x\) to both sides to get all \(x\)-terms on one side:
\[
-6x + 8x + 14 = 1
\]
\[
2x + 14 = 1
\]

2. Subtract 14 from both sides to isolate the term with \(x\):
\[
2x + 14 - 14 = 1 - 14
\]
\[
2x = -13
\]

3. Divide both sides by 2 to solve for \(x\):
\[
x = \frac{-13}{2}
\]

Solution for Equation 10:
\[
\boxed{x = -\frac{13}{2}}
\]

---

Final Answers:

\[
\boxed{
\begin{aligned}
1. & \quad n = -8 \\
2. & \quad p = -\frac{16}{3} \\
3. & \quad n = -\frac{9}{17} \\
4. & \quad p = -2 \\
5. & \quad x = 2 \\
6. & \quad v = \frac{85}{63} \\
7. & \quad x = -\frac{13}{7} \\
8. & \quad r = \frac{20}{9} \\
9. & \quad q = \frac{5}{16} \\
10. & \quad x = -\frac{13}{2}
\end{aligned}
}
\]