https://file.aiquickdraw.com/mathbot/file/17266204053259ek75jcz.jpg

Let's go through and solve each equation one by one from the image you provided. I’ll also determine how many solutions each equation has, as requested.

### 15. \( 4h + 5 = 9h \)
\[
4h + 5 = 9h
\]
Subtract \(4h\) from both sides:
\[
5 = 5h
\]
Divide by 5:
\[
h = 1
\]
**Solution**: \( h = 1 \) (one solution)

---

### 16. \( 2(3x - 6) = 3(2x - 4) \)
Expand both sides:
\[
6x - 12 = 6x - 12
\]
Subtract \(6x\) from both sides:
\[
-12 = -12
\]
This is always true, so the equation has **infinitely many solutions**.

---

### 17. \( 7t = 80 + 9t \)
\[
7t = 80 + 9t
\]
Subtract \(9t\) from both sides:
\[
-2t = 80
\]
Divide by \(-2\):
\[
t = -40
\]
**Solution**: \( t = -40 \) (one solution)

---

### 18. \( m + 3m = 4 \)
Simplify the left-hand side:
\[
4m = 4
\]
Divide by 4:
\[
m = 1
\]
**Solution**: \( m = 1 \) (one solution)

---

### 19. \( -b + 4b = 8b - b \)
Simplify both sides:
\[
3b = 7b
\]
Subtract \(3b\) from both sides:
\[
0 = 4b
\]
Divide by 4:
\[
b = 0
\]
**Solution**: \( b = 0 \) (one solution)

---

### 20. \( 6p + 1 = 3(2p + 1) \)
Expand the right-hand side:
\[
6p + 1 = 6p + 3
\]
Subtract \(6p\) from both sides:
\[
1 = 3
\]
This is never true, so the equation has **no solutions**.

---

### 21. \( 10z - 5 + 3z = 8 - z \)
Combine like terms:
\[
13z - 5 = 8 - z
\]
Add \(z\) to both sides:
\[
14z - 5 = 8
\]
Add 5 to both sides:
\[
14z = 13
\]
Divide by 14:
\[
z = \frac{13}{14}
\]
**Solution**: \( z = \frac{13}{14} \) (one solution)

---

### 22. \( 3(g - 1) + 7 = 3g + 4 \)
Expand both sides:
\[
3g - 3 + 7 = 3g + 4
\]
Simplify:
\[
3g + 4 = 3g + 4
\]
This is always true, so the equation has **infinitely many solutions**.

---

### 23. \( 17 - 20q = (-13 - 5q)4 \)
First, expand the right-hand side:
\[
17 - 20q = -52 - 20q
\]
Add \(20q\) to both sides:
\[
17 = -52
\]
This is never true, so the equation has **no solutions**.

---

### Summary:
1. **15**: \( h = 1 \) (one solution)
2. **16**: Infinitely many solutions
3. **17**: \( t = -40 \) (one solution)
4. **18**: \( m = 1 \) (one solution)
5. **19**: \( b = 0 \) (one solution)
6. **20**: No solution
7. **21**: \( z = \frac{13}{14} \) (one solution)
8. **22**: Infinitely many solutions
9. **23**: No solution

Would you like more details on any of these solutions?

### Related Questions:
1. How do you determine if an equation has no solution?
2. What happens when both sides of an equation simplify to the same value?
3. Can an equation have exactly two solutions?
4. What is the method for solving linear equations with variables on both sides?
5. How does expanding brackets work in solving equations?

### Tip:
When solving equations, always simplify both sides as much as possible before isolating the variable. This will help identify if there are no solutions or infinitely many solutions early on.

Solve each equation. Write how many solutions it has.

This guide provides step-by-step solutions for solving linear equations with variables on both sides. Includes how to determine if an equation has one solution, infinitely many solutions, or no solution at all. Suitable for students in Grades 7-9.

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