To find the values of p for which the given system of equations has a unique solution, we need to analyze the coefficients of x and y in both equations. The system of equations can be written as:

Equation 1: 4x - py + 8 = 0
Equation 2: 2x + 2y + 2 = 0


In order for the system of equations to have a unique solution, the two lines represented by the equations must intersect at a single point. This means that the slopes of the two lines must be different, i.e., the coefficients of x and y in the two equations must be different.

Let's compare the coefficients for x in both equations:
- In Equation 1, the coefficient of x is 4.
- In Equation 2, the coefficient of x is 2.

Since the coefficients are different, the slopes of the lines are different. Now, let's compare the coefficients for y in both equations:
- In Equation 1, the coefficient of y is -p.
- In Equation 2, the coefficient of y is 2.

If we want the lines to intersect at a single point, the slopes of the lines must be different. Therefore, the coefficients of y in both equations should also be different.


To find the values of p for which the system of equations has a unique solution, we need to set the coefficients of y in both equations to be different.

Let's equate the coefficients of y and solve for p:

-p ≠ 2

Simplifying the equation, we get:

p ≠ -2

This means that for any value of p except -2, the system of equations will have a unique solution. For p = -2, the system will not have a unique solution as the slopes of the two lines will be equal.


In conclusion, the values of p for which the given system of equations has a unique solution are all real numbers except -2. For any other value of p, the lines represented by the equations will intersect at a single point, resulting in a unique solution.