The thing that we did most, and that I remember most in the first semester is isolating the variable. The first time we did that, was when we tried to solve one-sided equations for a variable. For example, 4=5x-1. In that case x=1, and that can be found by reversing all arithmetical actions taken in the original equation until the variable is by itself. First, you would add one to both sides. The negative and positive ones would cancel each other out, and leave you with 4+1=5x. Then you would simplify the equation by combining like terms, or putting together the things that are alike, which would give you 5=5x. When two numbers or variables are right next to each other with a sign, the action taken is multiplication, and to undo multiplication, division is used. You would then be left with 1.

The next time we used "isolate the variable" was in two-sided equations. For example, 3x+1=2x+2. To solve that problem you would do the same, but need to subtract the variables as well. So, 2x would be subtracted from each side, along with 1. x=1 is what would be left, and that would be the answer.
 
 
When researching the price of Mountain Dew on the web, I found that a twelve pack of 12 oz cans of Mountain Dew (144 oz) costs around $5.50. I also found that a liter of Mountain Dew costs about $3. Given that a liter is 33.8 oz, I can determine that when buying by the liter, Mountain Dew costs about $3/33.8 oz. I can also determine that Mountain Dew by the 12 pack costs about $5.5/144 oz. Now I have rates of price, and I have to simplify and compare to find out which is the better deal.

$3 per 33.8 oz can be simplified to about $1 per 11 oz, and $5.5 per 144 oz can be simplified to $1 per 26 oz. While those numbers can be worked with from that point, I find it easier to first find a common denominator, by multiplying the fractions by the other fractions denominator over denominator. That way, 1/11 will be turned into 26/268 and 1/26 will be turned into 11/268. It is then clear to see that the 12 pack costs more per ounce, but it is also important to keep in mind that cans stay fresh longer, and other variables.