# Non-linear system with functions

$f:\mathbb R\to \mathbb R$ monotonically increasing.

Solve the system:

$$\begin{cases} f(x) + x = f(y) + y\\ x^2 + xy + y^2 = 12\end{cases}$$

Since $f$ is monotonically increasing and $f:\mathbb R\to \mathbb R$, isn't $f$ one-to-one?

But, I can't get it past that..

Can I replace $f(x)$ with $y$ so as to reduce the unknowns?

Since $f(x)+x$ is injective, as you already figured out, you can conclude that $x=y$. This makes the second equation trivial to solve.
You may not replace $f(x)$ with $y$, $y$ is a different, unrelated variable.
• You already used the first equation for figuring out that x=y is the only option. You basically said $f(x)+x=f(y)+y \rightarrow f(x)=f(y)\rightarrow x=y$. Aug 2 '15 at 12:20