Dear Uncle Colin,

I have the simultaneous equations $3x^2 - 3y = 0$ and $3y^2 - 3x = 0$. I’ve worked out that $x^2 = y$ and $y^2 = x$, but then I’m stuck!

- My Expertise Relatedto ((Hush, you.)) Simultaneous Equations? Not Nearly Enough!

Hi, MERSENNE, and thanks for your message!

There are a couple of ways I might recommend attacking this one: algebraically first, and graph first. Let’s do graph first, first.

### Graph first

The graph of $y=x^2$ is one you’re familiar with: a parabola that has a minimum at $(0,0)$.

$x = y^2$ is maybe less familiar, but it’s just the same as the one above, with the axes flipped - it’s a parabola opening to the right; at $(0,0)$, the tangent is vertical.

Even with the roughest of sketches, you can see that these intersect in two places: $(0,0)$ and $(1,1)$, and nowhere else.

### Algebra first

Alternatively, you can substitute $y = x^2$ into the equation $x = y^2$ to get $x = x^4$ - or, better yet, $x^4 - x = 0$.

This factorises as $x(x-1)(x^2 + x + 1) =0$, so the only real roots are when $x=0$ and $x=1$. (There are also two complex roots at $x = \frac{-1 \pm \sqrt{3}i}{2}$.)

Hope that helps!

- Uncle Colin