Question

I don't understand how to solve this linear quadratic system.

y^2 - 26 = -x^2
x-y = 6
If you're able to solve this, please explain it as well.

Answer 1

............................

Answer 2

the solutions are (1, - 5 ) and (5, - 1 )

y² - 26 = - x² → (1)

x - y = 6 → (2)

rearrange (2) in terms of x or y, that is

x = 6 + y

substitute this value into equation (1)

y² - 26 = - (6 + y )² ← distribute and rearrange into standard form

y² - 26 = - 36 - 12y - y²

2y² + 12y + 10 = 0 ← in standard quadratic form

divide through by 2

y² + 6y + 5 = 0

(y + 5)(y + 1 ) = 0 ← in factored form

y + 5 = 0 ⇒ y = - 5

y + 1 = 0 ⇒ y = - 1

substitute these values into x = 6 + y for corresponding values of x

y = - 5 : x = 6 - 5 = 1 ⇒ (1, - 5 ) is a solution

y = - 1 : x = 6 - 1 = 5 ⇒ (5, - 1 ) is a solution