To determine which equations have exactly one solution, we can solve each equation.
A) \(-5x + 12 = 5x - 5\)
Rearranging gives:
\(-5x - 5x = -5 - 12\)
\(-10x = -17\)
\(x = \frac{17}{10}\)
This equation has exactly one solution.
B) \(-5x + 12 = -5x - 12\)
Rearranging gives:
\(-5x + 5x = -12 - 12\)
\(0 = -24\)
This is a contradiction, so there are no solutions (not exactly one).
C) \(-5x + 12 = 5x + 12\)
Rearranging gives:
\(-5x - 5x = 12 - 12\)
\(-10x = 0\)
\(x = 0\)
This equation has exactly one solution.
D) \(-5x + 12 = -12x - 12\)
Rearranging gives:
\(-5x + 12 + 12 = -12x\)
\(12 + 12 = -12x + 5x\)
\(24 = -7x\)
\(x = -\frac{24}{7}\)
This equation has exactly one solution.
In conclusion, the equations that have exactly one solution are:
A, C, and D.