Question 970538
The symbol $ is defined by: a$b = a^2-4b^2     if x $y =3(y$x), express y in terms of x.
<pre>
{{{"x$y"}}}{{{""=""}}}{{{3("y$x")}}}

Involved in that equation are {{{"x$y"}}}, {{{"y$x"}}}, and {{{3("y$x")}}}.
We find those by substituting in {{{"a$b"}}}{{{""=""}}}{{{a^2-4b^2}}}

{{{"x$y"}}}{{{""=""}}}{{{x^2-4y^2}}}

{{{"y$x"}}}{{{""=""}}}{{{y^2-4x^2}}}

{{{3("y$x")}}}{{{""=""}}}{{{3(y^2-4x^2)}}}

So the given equation

{{{"x$y"}}}{{{""=""}}}{{{3("y$x")}}}

becomes

{{{x^2-4y^2}}}{{{""=""}}}{{{3(y^2-4x^2)}}}

We solve that for y in terms of x:

{{{x^2-4y^2}}}{{{""=""}}}{{{3y^2-12x^2}}}

{{{13x^2}}}{{{""=""}}}{{{7y^2}}}

{{{13x^2/7)}}}{{{""=""}}}{{{y^2}}}

{{{"" +- sqrt(13x^2/7)}}}{{{""=""}}}{{{y}}}
 
{{{y}}}{{{""=""}}}{{{"" +- sqrt(13x^2/7)}}}

{{{y}}}{{{""=""}}}{{{"" +- x*sqrt(13/7)}}}

That's the solution, but you may rationalize the denominator if you like:

{{{y}}}{{{""=""}}}{{{"" +- x*sqrt((13/7)*(7/7))}}}

{{{y}}}{{{""=""}}}{{{"" +- x*sqrt(91/49)}}}

{{{y}}}{{{""=""}}}{{{"" +- x*expr(sqrt(91)/7)}}}

Edwin</pre>