We are given the following system of equations:

1. \(x^2 - y^2 = 4\)

2. \(x + y = 2\)

We can solve this system using the method of substitution or elimination.

Let's solve using the method of substitution. From equation (2), we have \(y = 2 - x\). Substitute this value of \(y\) into equation (1):

\(x^2 - (2 - x)^2 = 4\)

Simplify the equation:

\(x^2 - (4 - 4x + x^2) = 4\)

Distribute the negative sign:

\(x^2 - 4 + 4x - x^2 = 4\)

Simplify further:

4x - 4 = 4

Now, solve for x:

4x = 8

x = 2

Now that we have x = 2, substitute it back into equation (2) to solve for y:

x + y = 2

2 + y = 2

y = 0

Therefore, the solutions are:

x = 2 and y = 0