We can solve this system of equations by substitution.

Step 1: Solve the first equation for x:
x = 1 - 2y

Step 2: Substitute x into the second equation:
(1 - 2y)^2 - 2(1 - 2y)y - y^2 = 1

Expand the left side:
1 - 4y + 4y^2 - 2 + 4y - y^2 - y^2 = 1

Combine like terms:
4y^2 - 3y - 2 = 1

Rearrange the equation:
4y^2 - 3y - 3 = 0

Now we have a quadratic equation in terms of y. Let's solve for y using the quadratic formula:
y = (-(-3) ± sqrt((-3)^2 - 4(4)(-3)) / 2(4)
y = (3 ± sqrt(9 + 48)) / 8
y = (3 ± sqrt(57)) / 8

Therefore, the solutions for y are:
y = (3 + sqrt(57)) / 8
y = (3 - sqrt(57)) / 8

Now we can substitute these values back into the first equation to solve for x.