One of my peers from high school asked me how can this problem be solved:
$$\begin{cases} x^2+y^2=z \\ x+y+z=m \end{cases}$$
Considering the mentioned equations, find $m$ such that the system has a unique solution.
An aspect regarding a numerical method came into my mind, given a system of nonlinear equations, if $\| J\| <1$ (where $J$ denotes the Jacobian) the system has a fixed point, hence a unique solution.
But I can't really use this for a 12th grader, so I'm kinda clueless on how other I can approach this problem.
Can you suggest another solution?
The second equation can be written as $z = m-x-y$.
Substituting this into the first equation gives $x^2+y^2 = m-x-y$
Move everything to the right side and complete the square:
$x^2+x+\dfrac{1}{4}+y^2+y+\dfrac{1}{4} = m+\dfrac{1}{2}$
$\left(x+\dfrac{1}{2}\right)^2+\left(y+\dfrac{1}{2}\right)^2 = m+\dfrac{1}{2}$
Now it should be easy to determine which value of $m$ yields only one solution.