You are given y in terms of x, so directly substitute:
Continue and solve for x.
y = 2x - 6
x^2 - xy + 2y^2 = 16
=> x^2 - x (2x-6) + 2(2x-6)^2 = 16
x^2 - 2 x^2 + 6x + 2( 4x^2 -12x + 36) = 16
-x^2 + 6x + 8x^2 -24x + 72 -16 = 0
7x^2 - 18x + 56=0
use the quadraic eqn. to solve for x, then when u get the two x values, plug it into y = 2x - 6 and u the two y values