Simultaneous Equations

1,
x = y +1
x^2 + y^2 = 5

(y+1)^2 + y^2 = 5
y^2 + 2y + 1 + y^2 = 5
2 y^2 + 2y -4 = 0
y^2 + y - 2 = 0
(y-1)(y+2)=0
y = 1 or y = -2

Sub y = 1 into x = y+1, x = 2.
Sub y = -2 into x = y+1, x =-1

So, x=2 ,y=1 or x=-1 ,y=-2

2,
y = 2x-1
x^2 - 2 y^2 = -1

x^2 - 2 (2x-1)^2 = -1
x^2 - 2(4 x^2 - 4x + 1) = -1
x^2 - 8 x^2 + 8x -2 = -1
x^2 - 8 x^2 + 8x = 1
- 7 x^2 + 8x -1= 0
7 x^2 - 8x +1= 0

(x-1)(x-1/7)=0
x = 1 or x = 1/7

Sub x =1 into y = 2x-1, y = 1.
Sub x = 1/7 into y = 2x-1, y = -5/7

So, x=1, y=1 or x=1/7, y = -5/7