Answer
(-1, 2) or (4, -3)
Work Step by Step
x + y = 1 can be written as y = 1 - x. Plug this into $x^{2} + xy - y^{2} = -5$
$x^{2} + x(1 - x) - (1 - x)^{2} = -5$
$x^{2} + x - x^{2} - 1 + 2x - x^{2} = -5$
$-x^{2} + 3x + 4 = 0$
Factor
(x + 1)(-x + 4) = 0
x + 1 = 0 or -x + 4 = 0
x = -1 or 4
Plug the x-values into y = 1 - x
y = 1 + 1 = 2
y = 1 - 4 = -3
(-1, 2) or (4, -3)