To find the value of \( k \) that is a solution to the equation \( -x^2 + 5x = -x + 5 \), we can first rearrange the equation:
\[ -x^2 + 5x + x - 5 = 0 \]
This simplifies to:
\[ -x^2 + 6x - 5 = 0 \]
Next, we can multiply through by \(-1\) to make it easier to solve:
\[ x^2 - 6x + 5 = 0 \]
Now we can factor the quadratic:
\[ (x - 1)(x - 5) = 0 \]
Setting each factor equal to zero gives us the solutions:
\[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \] \[ x - 5 = 0 \quad \Rightarrow \quad x = 5 \]
Next, we look for these values (1 and 5) in the second table where \( k \) is located:
\[ \begin{array}{c|c} x & -x + 5 \ \hline -1 & 6 \ k & 4 \ 3 & 2 \ 5 & 0 \ 7 & -2 \ \end{array} \]
From the second table, we see that \( k \) corresponds to the output for \( x \). Since our solutions to the equation are \( x = 1 \) and \( x = 5 \), we notice that:
- The value \( 5 \) is given (resulting in \( 0 \)).
- The value \( k \) which results in \( 4 \) must correspond to \( x = 1 \).
Thus, the solution to the equation \( -x^2 + 5x = -x + 5 \) implies:
The correct value of \( k \) is:
\[ \boxed{4} \]