Steven, first of all, exchange for all x's for y's.
y = (2ex- 3)/(7ex + 9) => x = (2ey - 3)/(7ey+ 9)
The next step is to multiply both sides for 7ey + 9; it's supposed to give you
x(7ey+ 9) = 2ey - 3
Apply now the distributive property for the left side of the equation.
7xey + 9x = 2ey - 3
Next, subtract 2eyon both sides.
7xey + 9x - 2ey= 2ey- 3 - 2ey
7xey + 9x - 2ey = -3
Next, subtract 9x on both sides.
7xey + 9x - 2ey - 9x = -3 - 9x
7xey - 2ey = -9x - 3
Having the terms with eyon one side of the equation, factor the left side using, precisely, ey as the common factor.
ey(7x - 2) = -9x - 3
Now divide each side by 7x - 2; the equation now must be solved for ey.
ey = (-9x - 3)/(7x - 2)
It is preferable to factor the numerator using -1 as the common factor. Note: Brackets instead of parentheses where needed in order to distinguish the numerator from the denominator.
ey = [-1(3 - 9x)]/(7x - 2)
Now, apply the -1 to the denominator.
ey = (3 - 9x)/(2 - 7x)
For the final step, you must get rid of the e in ey. For this to happen, you must use the natural logarithm (ln), which is some sort of an opposite operation for e. Note: The brackets are also used for this step, this time to distinguish the content of the natural logarithm.
y = ln[(3 - 9x)/(2 - 7x)]
Seeing that this equation is finally solved for y, you might say that this is the right answer, but, to make this result a lot more classy, let's use the property of logarithms concerning division, that is, log(a/b) = log(a) - log(b). Applying this law to the result, this gives you
y = ln(3 - 9x) - ln(2 - 7x)
This result is the inverse of the original exercise.