Solving A Non Linear System With a Cube

[B18]

Homework Statement

Find all solutions for the following system:
X^2-xy+2y^2=8
X^3-xy^2=0


2. Questions
What steps could I take to solve this? I've tried just about everything I can think of. I don't
know if I should solve the cubic equation for a variable and plug it in or what. I am petty stumped here guys.

The Attempt at a Solution

I took an x out of the x^3-xy^2 making it x(x^2-y^2)=0 and that is basically the only logical thing I have thought of. Thanks for any of your help!

 

[grindfreak]

There are a couple of ways to go about this. You're on the right track with the second equation, just factor x^2 - y^2 and you'll be able to find three solutions to try in the first equation. Plug them into the first equation and solve for y. All in all you should get 4 solutions for y and 5 for x, making the total number of (x,y) pairs = 5*4 = 20

 

[B18]

Ok so when I factor x^3-xy^2=0 I will get x=0 and x^2-y^2=0 which then results in x=±y. How else could I get answers for that. Does x=±y count as two answers? So my three answers from that second equation would be→x=0, x=y, x=-y.

 

[B18]

I came up with (0,2), (0,-2), (-y,2), (-y,2), (y,2), (y,-2). And when I put y= 2 in I end up getting the same first two answers.

 

[Ray Vickson]

Ok so when I factor x^3-xy^2=0 I will get x=0 and x^2-y^2=0 which then results in x=±y. How else could I get answers for that. Does x=±y count as two answers? So my three answers from that second equation would be→x=0, x=y, x=-y.

Yes, of course y = ± x is two solutions; after all, they say two different things.

RGV

 

[B18]

I agree, thanks for your help guys. I am still struggling a bit but I will try some more.

 

[B18]

Does anyone else have another way about this I am not getting as many answers as I expect.

 

[grindfreak]

Well you've already found three solutions from the second equation(x = 0, x=-y and x=y), what happens when you plug these into the first equation and solve?

 

[B18]

I end up with (0,2) (0,-2) and for the x= y I get some unfactorable stuff. I got (-y, root 2) and (-y, - root 2) I don't think that is correct tho but that is what I get hen substituting in -y for x in the first equation.