x^2 + y^2 = 25
y = x - 1
That's the inersection of a circle of radius 5, centered at the origin, with the straight line y = x-1.
There are two solutions.
Solve the two equations simultaneously. First, get a new equation in x or y only.
x^2 + y^2 = 25
-y^2 +x^2 -2x = -1
2x^2 -2x = 24
x^2 -x -12 = 0
(x +3)(x-4) = 0
x = -3 or 4
y = -4 (when x = -3) or 3 (when x=4)
(x,y) = (-3,-4) or (4,3)