To find the value of \( k \) that is a solution to the equation \( -x^2 + 5x = -x + 5 \), we can first simplify the equation.
Rearranging gives us:
\[ -x^2 + 5x + x - 5 = 0 \]
\[ -x^2 + 6x - 5 = 0 \]
Multiplying through by -1 to make it easier to solve:
\[ x^2 - 6x + 5 = 0 \]
Next, we can factor this equation:
\[ (x - 1)(x - 5) = 0 \]
This gives us the solutions:
\[ x = 1 \quad \text{and} \quad x = 5 \]
Now, we must check the table to determine which of the values correspond to \( k \).
In the table, we have:
- For \( x = 1 \), \( -x^2 + 5x = 4 \) and \( -x + 5 = 4 \).
- For \( x = 5 \), \( -x^2 + 5x = 0 \) and \( -x + 5 = 0 \).
- The table indicates that \( k \) is listed as \( 4 \) when \( x = 1 \).
The only value of \( k \) from the table that satisfies the equation \( -x^2 + 5x = -x + 5 \) is when \( x = 1\) which is indeed \( k = 4\).
Thus, the answer is:
4