System of equations with 2 parameters

[Hivoyer]

Homework Statement

I have a system of two equations:

3*x^2 - x + 3*y^2 = 0
2*x^2 - y + 2*y^2 = 0

Homework Equations

The Attempt at a Solution

I don't know how to express one with the other.I mean I can either have x = 3*y^2 + 3*x^2 or y = y = -2*y^2 - 2*x^2 and in both cases it becomes an utter mess.What can I do?

 

[mfb]

I would modify the second equation to get x^2 = ... and therefore x = ... and use this in the first equation.

 

[LCKurtz]

Here's another suggestion. You can see by inspection that (0,0) is one intersection point. Since both equations represent circles, put them in standard form and locate their centers. Determine the slope $m_1$ of the line of centers. The slope of the common chord between their intersection points will be $m_2=-\frac 1 {m_1}$. The line through the origin with that slope $m_2$ whose equation is $y=m_2x$ will pass through the other intersection point. Solve that with one of your circles. It works out pretty easily.

 

[haruspex]

One more suggestion: can you spot a multiplier that makes the quadratic terms in one equation the same as those in the other?