Question

x²+y²=34
x+y=2

Answer 1

from the 2nd: y = 2-x
plug into the first:
x^2 + (2-x)^2 = 34
x^2 + 4 - 4x + x^2 = 34
2x^2 - 4x - 30 = 0
x^2 - 2x - 15 = 0
(x - 5)(x + 3) = 0
x = 5 or x = -3
if x = 5, then y = -3
if x = -3, then y = 5

another way:
square the 2nd
x^2 + 2xy + y^2 = 4
34 + 2xy = 4
xy = -15

now we are looking for two numbers which have a sum of 2
and a product of -15,
Doesn't 5 and -3 come to mind?

similarly by just common sense:
The first few perfect squares are 1, 4, 9, 16, 25, 36
I don't have to go any further since the sum of two of them is only 34
I notice that 25 + 9 = 34 ===> 5^2 + 3^2 = 34
but 5+3 = 8, not 2, ahha, what if one of them is negative???
so 5 and -3 would work