To determine which equations have exactly one solution, let's analyze each option. An equation has exactly one solution when it can be simplified to a linear equation with one variable that can be isolated.
A. \( 6m - 9 = 6m \)
This simplifies to \( -9 = 0 \), which is a contradiction. Thus, it has no solutions.
B. \( 24a - 22 = -4(1 - 6a) \)
Expanding the equation gives:
\[ 24a - 22 = -4 + 24a \]
Subtracting \( 24a \) from both sides gives:
\[ -22 = -4 \], which is also a contradiction. Thus, it has no solutions.
C. \( 5n + 34 = -2(1 - 7n) \)
Expanding the right side gives:
\[ 5n + 34 = -2 + 14n \]
Rearranging gives:
\[ 5n - 14n = -2 - 34 \]
\[ -9n = -36 \]
Solving for \( n \) gives:
\[ n = 4 \]. So, it has exactly one solution.
D. \( 3n - 5 = -8(6 + 5n) \)
Expanding the right side gives:
\[ 3n - 5 = -48 - 40n \]
Rearranging gives:
\[ 3n + 40n = -48 + 5 \]
\[ 43n = -43 \]
Solving for \( n \) gives:
\[ n = -1 \]. So, it has exactly one solution.
E. \( p - 6 = -8 + p \)
Subtracting \( p \) from both sides gives:
\[ -6 = -8 \], which is also a contradiction. So, it has no solutions.
Based on the analysis, the equations with exactly one solution are:
C. \( 5n + 34 = -2(1 - 7n) \)
D. \( 3n - 5 = -8(6 + 5n) \)