Starting system:

y - x = 1
y = x² - 2x - 9

In this example, we use the simpler "substitution" method.
Solve the top equation for "y", to get:

y = x + 1

Substitute "x + 1" for "y" in the 2nd equation, to get :

x + 1 = x² - 2x - 9
or
x² - 3x - 10 = 0.

Now factor the left side, to get :
(x - 5)(x + 2) = 0

Thus, either x = 5 and hence y = 6
or x = -2 and hence y = -1