[Grade 9] Rearranging Equations - Changed schools and there is an obvious gap in my knowledge but I just don't get the process :(

Hello,

I recently changed schools after moving across the country. It is very clear to me that I have some major gaps in my math skills. I am confident I can catch up on most of it but rearranging equations is killing me. The process doesn't make sense - and every time I think I'm doing the right thing (or something that makes sense to me) it turns out that I am not.

For example I was working on the following:

6a + 4 = 5 - 2a

The teacher's example went as follows:

6a = 1 - 2a (getting rid of the 5 by moving to other side)

8a = 1 (getting rid of 2a by moving to the other side)

a = 1/8 (or 0.125 - solving for 8 by dividing both sides by 8)

OK, fine. I get that having it shown to me. But... and this is so embarrassing... my first effort was as follows o_O'

6a + 4 = 5 - 2a

6a + 4 (-5) = 5 - 5 - 2a (subtracting 5 from both sides to get 2a alone)

6a + 4 (-5) = -2a

6a + (-1) = - 2a

(-1) = -2a - 6a (subtracting 6a from both sides)

-1 = -4a (negative 2 minus 6)

-1/-4 = -4a/-4 (dividing both sides by -4 to isolate a)

a = -1/-4 (or 0.25)

Alright, obviously I'm doing something terribly wrong because my effort looked NOTHING like the teacher's example but I don't know the rules. I understand the concept: "whatever happens to one side of the equation has to happen to the other side" but how do you know what gets moved?

• why did the teacher move the 4 to the right side instead of the 5 to the left side?
• why did the move 2a to the right instead of 6a to the left?

Are there rules for this? I've looked online and all I can find are examples of things I understand, like rearranging simple science equations such as solving for density.

I've spent a fair amount of time on this and I'm starting to feel like an idiot because I just don't see the rules I'm supposed to follow - or maybe there's some other glaring problem with my math that I'm not aware of.

Any help is appreciated and thanks in advance!