Given Equations:
The given equations are:
1) x² y² - 25 = 0
2) x - y - 1 = 0

Step 1: Solving the second equation for x
We can rewrite the second equation as:
x = y + 1

Step 2: Substituting the value of x in the first equation
Substituting the value of x from the second equation into the first equation, we get:
(y + 1)² y² - 25 = 0

Step 3: Expanding and simplifying
Expanding the equation, we have:
(y² + 2y + 1) y² - 25 = 0

Simplifying further, we get:
y^4 + 2y^3 + y^2 - 25 = 0

Step 4: Factoring the equation
We can factor the equation as follows:
(y^2 + 5) (y^2 - 5) = 0

Step 5: Solving for y
Setting each factor equal to zero, we have:
y^2 + 5 = 0 or y^2 - 5 = 0

Solving the first equation, we get:
y^2 = -5
Taking the square root of both sides, we have:
y = ±√(-5)

Since the square root of a negative number is not a real number, there are no real solutions for y.

Step 6: Finding the corresponding values of x
Since there are no real solutions for y, there are no corresponding values for x.

Conclusion:
Therefore, the given system of equations does not have any real solutions.