solve algebraically

x^2 + y^2 = 16
x - y = 4

from the second, x = y+4

sub that into the first

(y+4)^2 + y^2 = 16
y^2 + 8y + 16 + y^2 = 16
2y^2 + 8y = 0
y^2 + 4y = 0
y(y+4) = 0
y = 0 or y = -4

in x = y+4
1. if y = 0, then x = 4
2. if y = -4 then y = 0

so we have (4,0) and (0,-4) as the solution