money9111 wrote:I'm getting better at recognizing when equations are equal, but wanted to know if anyone went through a mental process to figure it out.

Example:

Determine whether it is possible to solve for x using the given equations (do not solve).

3a+2b+x=8 and 12a+8b+2x=4. So I immediately looked to the right of the equal sign to see if by dividing the 1st equation by 2 would result in the 2nd one and this was not the case...

If you'll be able to solve for x, you'll need to eliminate the other unknowns. In the most typical situations, we do this by making the numbers in front of a (or b, as you like) equal in both equations. So if we have:

eqn 1: 3a+2b+x=8
eqn 2: 12a+8b+2x=4

we can multiply the first equation by 4 to equalize the numbers in front of a:

eqn 1: 12a + 8b + 4x = 32
eqn 2: 12a + 8b + 2x = 4

Now we can see that, by subtracting the second equation from the first, not only will the a's disappear but so will the b's, and we can find x.

That won't always happen of course; sometimes you can only eliminate one of the two letters.