Answer
(0, 1) or (4, -3)
Work Step by Step
x + y = 1 can be written by y = 1 - x. Plug this into $(x - 1)^{2} + (y + 2)^{2} = 10$
$(x - 1)^{2} + (1 - x + 2)^{2} = 10$
$(x - 1)^{2} + (- x + 3)^{2} = 10$
$x^{2} - 2x + 1 + x^{2} - 6x + 9 = 10$
$2x^{2} - 8x = 0$
Factor
2x(x - 4)= 0
2x = 0 or x - 4 = 0
x = 0 or 4
Plug x-values into y = 1 - x
y = 1 - 0 = 1
y = 1 - 4 = -3
(0, 1) or (4, -3)