Question

which of the following has exactly one solution?
-3x + 8 = -3x - 8
-3x - 8 = 3x - 8
3x - 8 = 3x + 8
-8x + 3 = -8x + 3

Answer 1

To determine which of these equations has exactly one solution, let's analyze each one.

1. **Equation:** \(-3x + 8 = -3x - 8\)
**Simplification:** Add \(3x\) to both sides:
\(8 = -8\)
This is a false statement (8 does not equal -8).
**Conclusion:** No solutions.

2. **Equation:** \(-3x - 8 = 3x - 8\)
**Simplification:** Add \(3x\) to both sides:
\(-8 = 6x - 8\)
Add 8 to both sides:
\(0 = 6x\)
Divide by 6:
\(x = 0\)
**Conclusion:** One solution (\(x = 0\)).

3. **Equation:** \(3x - 8 = 3x + 8\)
**Simplification:** Subtract \(3x\) from both sides:
\(-8 = 8\)
This is a false statement (−8 does not equal 8).
**Conclusion:** No solutions.

4. **Equation:** \(-8x + 3 = -8x + 3\)
This is an identity, meaning it is true for all values of \(x\).
**Conclusion:** Infinite solutions.

### Final Conclusion:
The equation that has exactly **one solution** is:
\(-3x - 8 = 3x - 8\) (with the solution \(x = 0\)).