Question

How do you solve the system `x^2+y^2=81` and `x+y=0`?

Answer 1

`(9/sqrt2,-9/sqrt2)` or `(-9/sqrt2,9/sqrt2)`

Explanation:

Start with the simpler equation `x+y=0`.

You can rewrite this to be `y=-x` (or `x=-y`).

Now, replace all the `y`s in the more complicated equation with `-x`s.

`x^2+y^2=81rArrx^2+(-x)^2=81`.

`x^2+(-x)^2=81rArrx^2+x^2=81`.

`x^2+x^2=81rArr2x^2=81`.

`2x^2=81rArrx^2=81/2rArrx=+-9/sqrt2`.

Now use the equation from before `y=-x` to see that the solutions are the ordered pairs `(9/sqrt2,-9/sqrt2)` or `(-9/sqrt2,9/sqrt2)`.

You can also see this by graphing both equations and seeing their points of intersection.

graph{(x+y)(x^2+y^2-81)=0 [-40, 40, -20, 20]}