To solve this system of equations, we can set the two equations equal to each other since they both equal to y:

x^2 - 8 = x - 6

Next, we can rearrange the equation to set it equal to zero:

x^2 - x - 2 = 0

Now we can factor the quadratic equation:

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

Setting each factor equal to zero gives us two possible solutions for x:

x - 2 = 0 or x + 1 = 0

x = 2 or x = -1

Now that we have found the values of x, we can substitute them back into one of the original equations to find the corresponding values of y:

For x = 2:
y = 2 - 6
y = -4

For x = -1:
y = (-1)^2 - 8
y = 1 - 8
y = -7

Therefore, the solutions to the system of equations are x = 2, y = -4 and x = -1, y = -7.