Hi, guys. We are learning about an an algebraic proportional equality (this is not what it is called, just a description of sorts) something in class right now, and I haven't been able to comprehend it. I asked my professor what it was called and he said it didn't really have a name and that it was just an application of different algebraic principles on ratios. So I don't know what to look for specifically online to discover comprehension on my own. My best idea thus far is to post an example here and see if you guys can point me in the right direction as to how to learn it. It just hasn't clicked with me yet, as it is a different way to think about algebra.

So, here is an example (well, I will post a couple, one considered hard and one considered easy):

Given,

if you to what would you to to to make both sides equal?

And another:

Given,

what would you do to to make both sides equal if we changed to ? What about ?

I'm not looking for the answers here. I'm looking to understand. I know that you can't change the variables themselves (as in giving it an exponent), you have to add an additional coefficient beside them.

Any help?