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 actually multiplied one term on the right-hand side by $\;1 = \frac 22.\;$ That's valid! $\;1\cdot x = x,\;$ whatever the value of $\,x\,$. In doing so, you haven't changed the value of the original fraction $10/3$, nor have you changed the quantity on either side of the equation. $$\dfrac{10}{3} = \dfrac {10}{3}\cdot \dfrac 22 = \frac {20}{6}$$
$$y = -\frac 56 - \frac{10}6 + \frac{10}{3} = -\frac 56 - \frac{10}6 + \frac{20}{6}$$
You are correct, though, that it would not be valid to multiply the fraction $10/3$ by $2$: resulting in $2 \cdot \dfrac {10}{3} = \dfrac {20}3,\;$ unless you also multiplied $y$ by $2$, and multiplied each of the other terms in the sum on the right-hand side of the equation by $2$.
You did not multiple it by two but $\frac{2}{2}=1$ instead.