Math worksheet for solving equations with different solution types, featuring 12 problems for practice.

Problem: Solving Equations of Different Solution Types

We are tasked with solving each equation and determining whether it has one solution, infinite solutions, or no solutions. Let's solve each equation step by step.

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#### Part 1: Solve each equation

##### 1) \( -68 + 30k = 10(3k - 7) + 2 \)

1. Distribute the 10 on the right-hand side:
\[
-68 + 30k = 10 \cdot 3k - 10 \cdot 7 + 2
\]
\[
-68 + 30k = 30k - 70 + 2
\]

2. Simplify the right-hand side:
\[
-68 + 30k = 30k - 68
\]

3. Subtract \( 30k \) from both sides:
\[
-68 = -68
\]

4. This is a true statement, which means the equation is an identity. Therefore, it has infinite solutions.

\[
\boxed{\text{Infinite solutions}}
\]

##### 2) \( 48(m + 3) - 102 = 6(8m + 7) \)

1. Distribute on both sides:
\[
48m + 48 \cdot 3 - 102 = 6 \cdot 8m + 6 \cdot 7
\]
\[
48m + 144 - 102 = 48m + 42
\]

2. Simplify both sides:
\[
48m + 42 = 48m + 42
\]

3. Subtract \( 48m \) from both sides:
\[
42 = 42
\]

4. This is a true statement, which means the equation is an identity. Therefore, it has infinite solutions.

\[
\boxed{\text{Infinite solutions}}
\]

##### 3) \( 21 + 3h = 3(h + 7) \)

1. Distribute on the right-hand side:
\[
21 + 3h = 3h + 3 \cdot 7
\]
\[
21 + 3h = 3h + 21
\]

2. Subtract \( 3h \) from both sides:
\[
21 = 21
\]

3. This is a true statement, which means the equation is an identity. Therefore, it has infinite solutions.

\[
\boxed{\text{Infinite solutions}}
\]

##### 4) \( 4 = -8z - 7 + 8z \)

1. Combine like terms on the right-hand side:
\[
4 = (-8z + 8z) - 7
\]
\[
4 = 0 - 7
\]
\[
4 = -7
\]

2. This is a false statement, which means the equation has no solutions.

\[
\boxed{\text{No solutions}}
\]

##### 5) \( -8n + 7 + 8n = 7 \)

1. Combine like terms on the left-hand side:
\[
(-8n + 8n) + 7 = 7
\]
\[
0 + 7 = 7
\]
\[
7 = 7
\]

2. This is a true statement, which means the equation is an identity. Therefore, it has infinite solutions.

\[
\boxed{\text{Infinite solutions}}
\]

##### 6) \( 3 + 4v = 6(5 - 7v) \)

1. Distribute on the right-hand side:
\[
3 + 4v = 6 \cdot 5 - 6 \cdot 7v
\]
\[
3 + 4v = 30 - 42v
\]

2. Add \( 42v \) to both sides:
\[
3 + 4v + 42v = 30
\]
\[
3 + 46v = 30
\]

3. Subtract 3 from both sides:
\[
46v = 27
\]

4. Divide by 46:
\[
v = \frac{27}{46}
\]

5. This equation has one solution.

\[
\boxed{v = \frac{27}{46}}
\]

---

#### Part 2: State whether each equation has one, infinite, or no solutions. If just one solution, solve for the variable.

##### 7) \( 2(-5f - 9) = -22 - 10f \)

1. Distribute on the left-hand side:
\[
2 \cdot (-5f) + 2 \cdot (-9) = -22 - 10f
\]
\[
-10f - 18 = -22 - 10f
\]

2. Add \( 10f \) to both sides:
\[
-18 = -22
\]

3. This is a false statement, which means the equation has no solutions.

\[
\boxed{\text{No solutions}}
\]

##### 8) \( -66 = 9(-7 - 8r) + 72r \)

1. Distribute on the right-hand side:
\[
-66 = 9 \cdot (-7) + 9 \cdot (-8r) + 72r
\]
\[
-66 = -63 - 72r + 72r
\]

2. Combine like terms on the right-hand side:
\[
-66 = -63 + 0
\]
\[
-66 = -63
\]

3. This is a false statement, which means the equation has no solutions.

\[
\boxed{\text{No solutions}}
\]

##### 9) \( 6(f - 10) = -60 + 6f \)

1. Distribute on the left-hand side:
\[
6f - 60 = -60 + 6f
\]

2. Subtract \( 6f \) from both sides:
\[
-60 = -60
\]

3. This is a true statement, which means the equation is an identity. Therefore, it has infinite solutions.

\[
\boxed{\text{Infinite solutions}}
\]

##### 10) \( 12 = -6(9t - 2) + 54t \)

1. Distribute on the right-hand side:
\[
12 = -6 \cdot 9t + (-6) \cdot (-2) + 54t
\]
\[
12 = -54t + 12 + 54t
\]

2. Combine like terms on the right-hand side:
\[
12 = 12
\]

3. This is a true statement, which means the equation is an identity. Therefore, it has infinite solutions.

\[
\boxed{\text{Infinite solutions}}
\]

##### 11) \( 8(5 - 4c) + 25c + 7c = 10 \)

1. Distribute on the left-hand side:
\[
8 \cdot 5 + 8 \cdot (-4c) + 25c + 7c = 10
\]
\[
40 - 32c + 25c + 7c = 10
\]

2. Combine like terms:
\[
40 + (-32c + 25c + 7c) = 10
\]
\[
40 + 0 = 10
\]
\[
40 = 10
\]

3. This is a false statement, which means the equation has no solutions.

\[
\boxed{\text{No solutions}}
\]

##### 12) \( 8 + u = 5u - 4(u + 9) \)

1. Distribute on the right-hand side:
\[
8 + u = 5u - 4u - 4 \cdot 9
\]
\[
8 + u = 5u - 4u - 36
\]

2. Simplify the right-hand side:
\[
8 + u = u - 36
\]

3. Subtract \( u \) from both sides:
\[
8 = -36
\]

4. This is a false statement, which means the equation has no solutions.

\[
\boxed{\text{No solutions}}
\]

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Final Answers:

1. Infinite solutions
2. Infinite solutions
3. Infinite solutions
4. No solutions
5. Infinite solutions
6. \( v = \frac{27}{46} \)
7. No solutions
8. No solutions
9. Infinite solutions
10. Infinite solutions
11. No solutions
12. No solutions

\[
\boxed{\text{See detailed solutions above}}
\]