Butting in myself, I extend lex’s post to show you why what he advises is very useful. In fact, the technique is commonplace in calculus. Why it is not taught in algebra is beyond my comprehension.
You have this messy equation
[MATH]10050 * 1.0725^N + 5000 * \dfrac{1}{0.0725} * (1.0725^n - 1) = 60000.[/MATH]
It will take time to manipulate that beast, and the opportunity for error is great. So SUBSTITUTE VARIABLES.
[MATH]\text {Set } x = 1.075^N.[/MATH]
[MATH]\therefore 10050x + \dfrac{5000}{0.0725} (x - 1) = 60000.[/MATH]
Already it looks less strange, looks easier to work with, and looks less error prone.
Frankly I would not proceed as your example does: I would first CLEAR FRACTIONS.
That is simple in this case. Just multiply both sides of the equation by 0.0725 which gives us this:
[MATH]728.625x + 5000(x - 1) = 4350 \implies 728.625x + 5000x - 5000 = 4350 \implies[/MATH]
[MATH]5728.625x = 4350 + 5000 = 9350 \implies x = \dfrac{9350}{5728.625} \approx 1.632.[/MATH]
First year algebra. Distribution error is very unlikely. Now substitute back
[MATH]x \approx 1.632 \implies 1.0725^N \approx 1.632.[/MATH]
You will be told that substituting simple variables for expressions and clearing fractions are unnecessary mathematically. True. But if they make things quicker and less error prone, they can be SUPER helpful.
