Question

Which two problems have infinite solutions?

2x+3=2x+7
7x+3=7x+3
4x+22=8x+10
5x+5=5x+4
3x+18=5x+8
3(4x+8)=4(3x+6)

Answer 1

To determine which equations have infinite solutions, we need to analyze each equation:

1. **2x + 3 = 2x + 7**
- Subtract \(2x\) from both sides: \(3 = 7\). This is a false statement. No solutions.

2. **7x + 3 = 7x + 3**
- This simplifies to \(0 = 0\), which is true for all \(x\). Infinite solutions.

3. **4x + 22 = 8x + 10**
- Rearranging gives \(4x + 22 - 10 = 8x\) or \(12 = 4x\), which leads to \(x = 3\). One solution.

4. **5x + 5 = 5x + 4**
- Subtract \(5x\) from both sides: \(5 = 4\), which is false. No solutions.

5. **3x + 18 = 5x + 8**
- Rearranging gives \(3x - 5x = 8 - 18\) or \(-2x = -10\), which leads to \(x = 5\). One solution.

6. **3(4x + 8) = 4(3x + 6)**
- Expanding both sides: \(12x + 24 = 12x + 24\), which again simplifies to \(0 = 0\). True for all \(x\). Infinite solutions.

The two equations that have infinite solutions are:
- **7x + 3 = 7x + 3**
- **3(4x + 8) = 4(3x + 6)**