The non-negative real numbers and satisfy the two equations
x^2+y^2+z^2 &= 9\\
xy+yz+zx &= 8.
What is the sum of $x,y,$ and $z?$
This is kind of a guess and check answer.
The only way this could work is if x y and z equal 3, 0, 0 in some order, or 1, 2, 2. Since 3, 0, 0 isn't very good for the second equation, we'll go with 1, 2, 2. When you plug it in we get 1+4+4=9 for the first equation and 2+4+2=8 for the second equation. So we get the sum is 5