The equation 2x + 4k − 9 = kx − k + 1 is really a family of equations, because for each value of k, we get a different equation with the unknown x. the letter k is called a parameter for this family. what value should we pick for k to make the given value of x a solution of the resulting equation? (if all real numbers are solutions, enter reals.)

You haven't shared "the given value of x," or, if you have, you haven't drawn attention to it.

Just suppose we were to choose x = 4 as a possible solution and then try to find a value of the parameter k that would make x = 4 an actual solution.

2(4) + 4k - 9 = (4)(4) - (4) + 1

Then 8 + 4k - 9 = 16 - 4 + 1, or       4k - 1 = 13.  Then 4k = 14, and k = 14/4, or (after reduction)  k = 7/2   

If the parameter k equals 7/2, then x = 4 is a solution to the given equation.

To check this out further, start with the proposed solution x = 5 and find k.