# How to simplify these equations into one

I had two equations with one variable (R) and two constants (A and B) in the form:

$$y = \frac{B(A^2-R^2)}{AB-R^2}$$

and

$$x = \frac{2ABR}{AB-R^2}$$

Now, I want to combine these equations and solve for y.

To do that I solved both equations for R and got:

$$R =\sqrt{\frac{ yAB-BA^2}{y-B}}$$

and

$$R =\frac{ -2AB \pm \sqrt{(AB)^2+ABx^2}}{x}$$

Combining these I got:

$$\sqrt{\frac{ yAB-BA^2}{y-B}} =\frac{ -2AB \pm \sqrt{(AB)^2+ABx^2}}{x}$$

The question then is how can I simplify this equation further to solve for y? (ie. to get it into the form y = . . . ). Am I at least on the right track?

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Hint: Did you notice that both denominators are the same? – Amzoti Sep 5 '13 at 4:37
Yes, of course. But I wasn't sure of how to use that information.. – Spherus Sep 5 '13 at 4:46
in your equation for R, I think it should be -AB instead of -2AB – Vikram Sep 5 '13 at 6:07
$y=\frac{x}{2A}[\frac{A^2}{R}-1]$ – Vikram Sep 5 '13 at 6:18

If you want to get rid of the variable $R$ as well, here's the lazy way to do it. First transform your equation from,

$$y = \frac{p}{q}$$

to the form,

$$qy-p = 0$$

Go to Wolframalpha and copy-paste this:

  Factor[Resultant[y(a b-r^2)-b(a^2-r^2), x(a b-r^2)-2a b r, r]]


Don't use capitals and put spaces between letters. This command eliminates the variable $r$ (or you could choose any variable.) The output is a quadratic in $y$. Solving for it, I get,

$$y=\frac{ab(a+b)\pm\sqrt{ab(a-b)(ab+x^2)}}{2ab}$$

Once you have $y$, you can recover $R$ (as you solved it) as,

$$R = \sqrt{\frac{ yAB-BA^2}{y-B}} =\frac{ -2AB \pm \sqrt{(AB)^2+ABx^2}}{x}$$

choosing the appropriate sign. (Note that $a,b = A,B$.)

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