Question

One Solution, No Solution, Infinite Solutions
Match to the correct expression

6(2x - 3) = 12(x - 3) + 18
4(3x - 3) = 12(x - 3) + 18
4(2x - 3) = 12(x - 3) + 18

Answer 1

To determine whether each equation has one solution, no solutions, or infinite solutions, we need to simplify each expression and compare both sides.

### Equation 1:
**6(2x - 3) = 12(x - 3) + 18**

1. Expand both sides:
- Left side: \(6(2x - 3) = 12x - 18\)
- Right side: \(12(x - 3) + 18 = 12x - 36 + 18 = 12x - 18\)

2. So we have:
\[
12x - 18 = 12x - 18
\]
This simplifies to \(0 = 0\), which is always true.

**Conclusion:** Infinite solutions.

---

### Equation 2:
**4(3x - 3) = 12(x - 3) + 18**

1. Expand both sides:
- Left side: \(4(3x - 3) = 12x - 12\)
- Right side: \(12(x - 3) + 18 = 12x - 36 + 18 = 12x - 18\)

2. So we have:
\[
12x - 12 = 12x - 18
\]
Subtract \(12x\) from both sides:
\[
-12 = -18
\]
This is not true.

**Conclusion:** No solutions.

---

### Equation 3:
**4(2x - 3) = 12(x - 3) + 18**

1. Expand both sides:
- Left side: \(4(2x - 3) = 8x - 12\)
- Right side: \(12(x - 3) + 18 = 12x - 36 + 18 = 12x - 18\)

2. So we have:
\[
8x - 12 = 12x - 18
\]
Rearranging gives:
\[
-12 + 18 = 12x - 8x
\]
\[
6 = 4x \implies x = \frac{6}{4} = \frac{3}{2}
\]

**Conclusion:** One solution.

---

### Summary of Solutions:
1. **6(2x - 3) = 12(x - 3) + 18**: **Infinite solutions**
2. **4(3x - 3) = 12(x - 3) + 18**: **No solutions**
3. **4(2x - 3) = 12(x - 3) + 18**: **One solution**