Question

How do you solve the system `9x^2+4y^2=36` and `-x+y=-4`?

Answer 1

Substitute `x - 4` for y in `9x^2 + 4y^2 = 36` to discover that there are no real roots for the resulting quadratic, therefore, the line does not intersect with the ellipse.

Explanation:

Given:
`y = x - 4`
`9x^2 + 4y^2 = 36`

`9x^2 + 4(x - 4)^2 = 36`

`9x^2 + 4(x^2 - 8x + 16) = 36`

`13x^2 - 32x + 64 = 36`

`13x^2 - 32x + 28 = 0`

`b^2 - 4(a)(c) = (-32)^2 - 4(13)(28) = -432`

There are no real roots, therefore, the line does not intersect with the ellipse