Question

How to solve this system of equal: x+y+z=1;x^2+y^2+z^2=1

Answer 1

There are infinitely many solutions.

Algebraically, we can eliminate $z$ and try to solve for $x,y$:

$x+y+z=1\implies z=1-x-y$

Then

$x^2+y^2+(1-x-y)^2=1$

$\implies x^2+y^2+1-2x+x^2-2y+y^2+2xy=1$

$\implies x^2-x+y^2-y+xy=0$

which is the equation of an ellipse.