To determine which of the given equations have exactly one solution, let’s analyze each equation step-by-step:
### Equation A: \( 2x - 31 = 2x - 31 \)
1. Subtract \( 2x \) from both sides:
[tex]\[
2x - 2x - 31 = 2x - 2x - 31
\][/tex]
2. Simplify:
[tex]\[
-31 = -31
\][/tex]
This is a true statement, meaning the equation holds for all values of \( x \). Hence, it has infinitely many solutions.
### Equation B: \( 2x - 31 = -2x - 31 \)
1. Add \( 2x \) to both sides:
[tex]\[
2x + 2x - 31 = -31
\][/tex]
2. Simplify:
[tex]\[
4x - 31 = -31
\][/tex]
3. Add 31 to both sides:
[tex]\[
4x - 31 + 31 = -31 + 31
\][/tex]
4. Simplify:
[tex]\[
4x = 0
\][/tex]
5. Divide by 4:
[tex]\[
x = 0
\][/tex]
This equation has exactly one solution, \( x = 0 \).
### Equation C: \( 2x + 31 = 2x - 31 \)
1. Subtract \( 2x \) from both sides:
[tex]\[
2x - 2x + 31 = 2x - 2x - 31
\][/tex]
2. Simplify:
[tex]\[
31 = -31
\][/tex]
This is a false statement, meaning the equation has no solutions.
### Equation D: \( 2x - 2 = 2x - 31 \)
1. Subtract \( 2x \) from both sides:
[tex]\[
2x - 2x - 2 = 2x - 2x - 31
\][/tex]
2. Simplify:
[tex]\[
-2 = -31
\][/tex]
This is a false statement, meaning the equation has no solutions.
From this analysis, the equation that has exactly one solution is:
[tex]\[ \text{Equation B: } 2x - 31 = -2x - 31 \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{2} \][/tex]
