Question

Which of the following equations has exactly one solution? (1 point)

- 3x + 8 = - 3x - 8

- 3x - 8 = 3x - 8

3x - 8 = 3x + 8

- 8x + 3 = - 8x + 3

Answer 1

To determine which equation has exactly one solution, let's solve each equation one at a time.

1. **Equation: \(3x + 8 = -3x - 8\)**
Adding \(3x\) to both sides:
\[
6x + 8 = -8
\]
Subtracting 8 from both sides:
\[
6x = -16
\]
Dividing by 6:
\[
x = -\frac{8}{3}
\]
**This equation has exactly one solution: \(x = -\frac{8}{3}\).**

2. **Equation: \(3x - 8 = 3x - 8\)**
If we subtract \(3x\) from both sides:
\[
-8 = -8
\]
This is a true statement but does not isolate \(x\).
**This equation has infinite solutions.**

3. **Equation: \(3x - 8 = 3x + 8\)**
Subtracting \(3x\) from both sides:
\[
-8 = 8
\]
This is a false statement and represents no solution.
**This equation has no solutions.**

4. **Equation: \(8x + 3 = -8x + 3\)**
Adding \(8x\) to both sides:
\[
16x + 3 = 3
\]
Subtracting 3 from both sides:
\[
16x = 0
\]
Dividing by 16:
\[
x = 0
\]
**This equation has exactly one solution: \(x = 0\).**

In conclusion, the equations that have exactly one solution are:

- \(3x + 8 = -3x - 8\) with the solution \(x = -\frac{8}{3}\)
- \(8x + 3 = -8x + 3\) with the solution \(x = 0\)

Therefore, the equations having exactly one solution are:
- **3x + 8 = -3x - 8** and **8x + 3 = -8x + 3**.