property of equality The property of equality says:
"The equals sign in an equation is like a scale: both sides, left and right, must be the same in order for the scale to stay in balance and the equation to be true."
So for example in the following equation, I want to isolate the x variable. So I cross-multiply both sides by 3/5:
5/3x  = 55
x = 3/5*55

What I did to one side, I had to to do the other.
However, take a look at the following problem:
y - 10/3 = -5/6(x + 2)
y = -5/6 - 10/6 + 10/3

So for we just use distributive property of multiplication to distribute -5/6 to the quantity of x + 2. Then since we isolate y, we add 10/3 to both sides.
However, now in order to add the two fractions, we find the greatest common factor is 6, so we multiple 10/3 by 2:
y = -5/6x - 10/6 + 20/6.

There is my question. Why can we multiply a term on the one side by 2, without having to do it on the other side?
After writing this question here, I'm now thinking that because 10/3 is equal to 20/6 we really didn't actually add anything new to the one side, and that's why we didn't have to add it to the other side. 

You did not multiple it by two but $\frac{2}{2}=1$ instead. 
