Find the coordinates of intersection for the following of line and curve

Question

x^2+2y^2=36
x+y=6

I know I hav to do it simultaneously but Im struggling

Please explain step by step so I know for future reference

Thanks

Answer 1

y = 6 - x
so
y^2 = 36 - 12 x + x^2
so
2 y^2 = 2 x^2 - 24 x + 72
so
3 x^2 - 24 x + 72 = 36

3 x^2 - 24 x + 36 = 0

x^2 - 8 x + 12 = 0

(x-2)(x-6) = 0

x = 2 or x = 6
x = 6 is trivial because then y = 0
if x = 2, y = 4
so
(6,0) and (2,4)