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.