SOLUTION: The symbol $ is defined by: a$b = a^2-4b^2 if x $y =3(y$x), express y in terms of x.

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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.
Answer by Edwin McCravy(20059) About Me  (Show Source):
You can put this solution on YOUR website!
The symbol $ is defined by: a$b = a^2-4b^2 if x $y =3(y$x), express y in terms of x.
%22x%24y%22%22%22=%22%223%28%22y%24x%22%29

Involved in that equation are %22x%24y%22, %22y%24x%22, and 3%28%22y%24x%22%29.
We find those by substituting in %22a%24b%22%22%22=%22%22a%5E2-4b%5E2

%22x%24y%22%22%22=%22%22x%5E2-4y%5E2

%22y%24x%22%22%22=%22%22y%5E2-4x%5E2

3%28%22y%24x%22%29%22%22=%22%223%28y%5E2-4x%5E2%29

So the given equation

%22x%24y%22%22%22=%22%223%28%22y%24x%22%29

becomes

x%5E2-4y%5E2%22%22=%22%223%28y%5E2-4x%5E2%29

We solve that for y in terms of x:

x%5E2-4y%5E2%22%22=%22%223y%5E2-12x%5E2

13x%5E2%22%22=%22%227y%5E2

13x%5E2%2F7%29%22%22=%22%22y%5E2

%22%22+%2B-+sqrt%2813x%5E2%2F7%29%22%22=%22%22y
 
y%22%22=%22%22%22%22+%2B-+sqrt%2813x%5E2%2F7%29

y%22%22=%22%22%22%22+%2B-+x%2Asqrt%2813%2F7%29

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

y%22%22=%22%22%22%22+%2B-+x%2Asqrt%28%2813%2F7%29%2A%287%2F7%29%29

y%22%22=%22%22%22%22+%2B-+x%2Asqrt%2891%2F49%29

y%22%22=%22%22%22%22+%2B-+x%2Aexpr%28sqrt%2891%29%2F7%29

Edwin