For anyone who saw this post in /r/InternetIsBeautiful, and is wondering why it is being downvoted to hell here, this is why:
This list of rules represents pretty much everything we hate about the way the general public perceives mathematics. The implication of this list seems to be "memorize these 25 rules and you'll know how to manipulate your way out of any algebra problem."
I promise you that no serious mathematician (or scientist, or engineer) thinks about algebraic manipulations as a list of rules. Far more important (and useful!) than these rules is a solid understanding of the operations at play. If you truly understand that subtraction is the same as adding a negative number, or division is the same as multiplying by the reciprocal, you eliminate the need for half of these rules. If you've internalized the notion of exponentiation as repeated multiplication it should come as absolutely no shock that a^m * a ^n = a^(m + n).
It is absolutely depressing to us that teachers across America present and teach algebra (and mathematics in general) in exactly this light. The author of this page can't even give a reasonable explanation for why a^0 = 1.
MTRE senior here, it'll stay difficult because it's fundamental to like 80% of the curriculum, from computing forces and vectors in your physics/ME classes, current and electro mumbo jumbo in your EE classes (way more than you might think), and then learning how to program math concepts like vectors in C++, MATLAB, etc..If you are passionate about becoming better at math then there is nothing to worry about, the online resources recommended here should be enough to help you if you are determined to learn.
Many students who continue to stumble through math courses, especially Calculus and Calc 2 have poor basic algebra skills. I would suggest heading to http://algebrarules.com/ for starters. Understanding these concepts fully is pivotal to future success. Do your homework, practice problems, and you will improve. Good luck.
You can follow the normal rules of algebra. Brushing up on these should help you. Just treat the variables as you would normal numbers.
> How do I know that a number is completely canceled out?
If you have one variable divided by itself, you can cancel it out in that fraction, but it may not be completely canceled out.
> How do I know where to start moving these values around ?
If you're trying to find one value, you want to isolate that. So if I had the equation Δx = V*i* + 1/2at^(2), and I want to isolate a, I would subtract V*i* from both sides, divide both sides by 1/2 and then divide both sides by t^(2).