finding solutions for a mixed variable equation

Hi there, this is a for a calculus question but it's more the algebra I'm having trouble with at the moment.

I've reduced an equation down to:

2y^2 + x^2 - 4xy + 4y - 4x = 0

and need to find all (x,y) pairs that make it true. Could someone show each step in solving this?

Thanks a lot

Re: finding solutions for a mixed variable equation

1) Did you encounter "Rotation of Axes" in your pre-calculus studies?

2) Without reference to the geometry, you can make a substitution that will solve the problem. $\displaystyle x = 2(u+v)$ and $\displaystyle y = \sqrt{2}(u-v)$ is one such substitution. You can then Complete the Square in u and in v and you'll ne much closer to a solution.

Re: finding solutions for a mixed variable equation

Hello, kiwi5!

Quote:

$\displaystyle \text{I've reduced an equation down to: }\:x^2 -4xy+ 2y^2 -4x + 4y \:=\: 0$

. . Are you sure this is correct?

$\displaystyle \text{and need to find }all\text{ pairs }(x,y)\text{ that make it true. }$ . ALL of them??

I'm quite certain that your equation is incorrect.

It represents a hyperbola . . . and hence has brizillions of solutions.

Re: finding solutions for a mixed variable equation

Quote:

Originally Posted by

**kiwi5** Hi there, this is a for a calculus question but it's more the algebra I'm having trouble with at the moment.

I've reduced an equation down to:

2y^2 + x^2 - 4xy + 4y - 4x = 0

and need to find all (x,y) pairs that make it true. Could someone show each step in solving this?

Thanks a lot

you can look at it as a quadratic in $\displaystyle x$. the equation becomes $\displaystyle x^2-(4y+4)x+(2y^2+4y)=0$. now apply the formula for solving a quadratic and you will have found $\displaystyle x$ in terms of $\displaystyle y$. that will do.