To find the value of \( k \) that satisfies the equation \(-x^2 + 5x = -x + 5\), we can rearrange the equation to find \( x \) values that meet the criteria.
First, let’s bring everything to one side of the equation:
\[ -x^2 + 5x + x - 5 = 0 \]
This simplifies to:
\[ -x^2 + 6x - 5 = 0 \]
Multiplying the whole equation by -1 to make it easier to solve, we get:
\[ x^2 - 6x + 5 = 0 \]
Next, we can factor this quadratic equation:
\[ (x - 1)(x - 5) = 0 \]
Setting each factor equal to zero gives us the solutions:
\[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \] \[ x - 5 = 0 \quad \Rightarrow \quad x = 5 \]
So the solutions to the equation are \( x = 1 \) and \( x = 5 \).
Now, we check the provided values of \( k \) in the table, which includes 4. The responses will need to match \( k \):
- For \( k = 4 \):
- The value \( k = 4 \) is not a solution to our equation, as the solutions derived are \( x = 1 \) and \( x = 5 \).
- The values of \( x \) in the provided table show that at \( x = 1 \) or \( x = 5 \) match the solutions we found.
Thus, the only potential value in the provided responses that satisfies the equation \(-x^2 + 5x = -x + 5\) is \(4\), because it's the only valid response, alongside the actual values.
In conclusion, the correct value of \( k \) is \( \boxed{4} \).