Equation of a plane which intersects the two surfaces?

The question asks to find the equation for a plane which intersects these two surfaces. I was sure that I solved it correctly, but my professor says it is wrong and I don't know what I am doing wrong here. Here are the two surfaces...

1) $\displaystyle x^2+2y^2-z^2+3x=1$

2) $\displaystyle 2x^2+4y^2-2z^2-5y=0$

**Here's what I did...**

I found three points that are on both surfaces, hence points at which they intersect. I did this by multiplying the first surface equation by two and subtracting the equations from each other to end up with it $\displaystyle 6x+5y=2$. After that, I just picked some numbers for x and y and then plugged it back into the surface equations to find z. The three points I found were $\displaystyle (1/3,0,1/3), (1/3,0,-1/3)$, and $\displaystyle (2,-2,\sqrt{17})$.

After finding these three points, I just used them to find the normal vector and then used the normal vector and the first point to find the equation. Here is the equation that I got...

$\displaystyle \frac{4}3x-\frac{10}9y+\frac{4}9=0$

Can someone please tell me whether this is correct or not. If its not correct, can someone tell me what I might be doing wrong? Thanks.