Can you solve this Maths question?

x = 2 + 2y :-

( 2 + 2y ) ² - 2y ( 2 + 2y ) + y² = 1
4 + 8y + 4y² - 4y - 4y² + y² = 1
y² + 4y + 3 = 0
(y + 3)(y + 1) = 0
y = - 3 , y = - 1
x = - 4 , x = 0

(- 4,- 3) , (0,- 1)

2nd equation (solve for x)
x - 2y = 2
x = 2y + 2

1st equation (plug x with 2y + 2, solve for y):
(2y + 2)² - (2y[2y + 2]) + y² = 1
4y² + 8y + 4 - 4y² - 4y + y² = 1
y² + 4y = - 3
y² + 2y = - 3 + 2²
y² + 2y = - 3 + 4
(y + 2y)² = 1
y + 2 = 1

1st set: y = 1 - 2, y = - 1
2nd set: y = - 1 - 2, y = - 3

Solve for x (2 sets of the 2nd equation):
1st set: x = 2(- 1) + 2, x = - 2 + 2, x = 0
2nd set: x = 2(- 3) + 2, x = - 6 + 2, x = - 4

Answer: 1st set: x = 0, y = - 1; 2nd set: x = - 4, y = - 3

Proof (1st set plugged to the 2nd equation):
0² - 2(0 * - 1) + (- 1)² = 1
0 - 0 + 1 = 1
1 = 1

Proof (2nd set plugged to the 2nd equation):
- 4² - 2(- 4 * - 3) + (- 3²) = 1
16 - 2(12) + (9) = 1
16 - 24 + 9 = 1
1 = 1

x^2 - 2xy + y^2 = 1 (solve by using substitution)
x - 2y = 2

x - 2y = 2
x = 2y + 2

x^2 - 2xy + y^2 = 1
(2y + 2)^2 - 2(2y + 2)y + y^2 = 1
(2y + 2)(2y + 2) - 2y(2y + 2) + y^2 - 1 = 0
4y^2 + 4y + 4y + 4 - 4y^2 - 4y + y^2 - 1 = 0
4y^2 - 4y^2 + y^2 + 8y - 4y + 4 - 1 = 0
y^2 + 4y + 3 = 0
y^2 + 3y + y + 3 = 0
(y^2 + 3y) + (y + 3) = 0
y(y + 3) + 1(y + 3) = 0
(y + 3)(y + 1) = 0

y + 3 = 0
y = -3

y + 1 = 0
y = -1

x - 2y = 2
x - 2(-3) = 2
x - (-6) = 2
x + 6 = 2
x = 2 - 6
x = -4

x - 2(-1) = 2
x - (-2) = 2
x + 2 = 2
x = 2 - 2
x = 0

∴ (x = -4 , y = -3) , (x = 0 , y = -2)

x^2 - 2xy + y^2 = 1 : (x - y)^2 = 1
x - 2y = 2 : x = 2(y+1)
substituting for x in eqn 1
( 2y +2 - y)^2 = 1
( y + 2)^2 = 1
y^2 +4 y +4 = 1
y^2 +4 y +3 = 0
(y + 3)( y +1) = 0
y = -3 : x = -4
y = -1 : x = 0

(x - y)² = 1

x - y = ± √1

x - y = 1
x - y = -1


Two new systems...

x - y = 1
x - 2y = 2

◊ subtract ◊

y = -1
x = 0

x - y = -1
x - 2y = 2

◊ subtract ◊

y = -3
x = -4


Two solutions (0, -1), (-4, -3)

x^2-2xy+y^2=1
(x-y)^2=1
so x-y=1 or x-y=-1

1st case:
x-y=1
x-2y=2

x=1+y
so, (1+y)-2y=2
y=-1
x=1-1=0

so (0,-1) is a solution

2nd case:
x-y=-1
x=-1+y
-1+y-2y=2
y=-3
x=-1-3=-4

so the solutions are (0,-1) and (-4,-3)