Question

How do you solve the system of equations `{(y =7 - x), (x^2 + y^2 = 65):}`?

Answer 1

There are two points that are solutions, `(8,-1) and (-1,8)`

Explanation:

Given:

`y^2+x^2=65 and y + x = 7`

Write the line as y as a function of x

`y^2+x^2=65 and y = 7-x`

Substitute `(7 -x)` for y into the circle:

`(7-x)^2+x^2=65`

Square first term:

`49-14x+x^2+x^2=65`

`2x^2-14x-16=0`

`x^2-7x-8=0`

`(x-8)(x+1)=0`

`x = 8 and x = -1`

Find the corresponding y values, using the equation, `y = 7-x`

y = -1 and y = 8

Answer 2

`(8, -1);(-1,8)`

Explanation:

From the second equation, we have `y = 7 - x`. Substituting:

`(7- x)^2 + x^2 = 65`

`49 - 14x + x^2 + x^2 = 65`

`2x^2 - 14x - 16 = 0`

`2(x^2 - 7x - 8) = 0`

`x^2 - 7x - 8 = 0`

`(x - 8)(x + 1) = 0`

`x = 8 and -1`

Since this system involves the equation of a circle (e.g `x^2 + y^2 = 65`), the solution set will be of the form `(a, b); (b, a)`.

The solution set therefore is `(8, -1);(-1, 8)`.

Hopefully this helps!