A way of talking about variables that seems to make sense to my students is by talking about them in what I call "groups." A group consists of a sign (+ or -), a number (sometimes 1, which is always invisible), and a variable (or letter). For example, -4x or +2a.
Then there is a special type of group that is just a sign and a number (my colleague Mr. Kelsey calls these Lonely Numbers, and Ive stolen that name), most people just know these just as numbers - they dont have a variable but you can treat them as a group. For example 13 or -8.To solve for a variable, follow these 3 steps:
1. Look for like terms to combine on either side of the = sign, without going across the = sign, and combine any that you can. Also if there is anything else, such as distributing, you can do on either side, do that first.
2. Next, combine any like terms across the = sign by moving the groups across. To move any group across the = sign, simply change the sign of it (+ or -) and put this on both sides of the = sign (this will cancel the group out on its original side; on the other side, combine like terms if you can). Do this until the group containing the variable you are solving for is by itself on one side of the = sign.
3. After the group with the variable you're solving for is by itself, then break it up by dividing by exactly what is being multiplied by the variable, you have to divide by this for every group on both sides of the = sign. This should leave the variable by itself (multiplied by 1).
Note: If a your variable is being divided by something, for example x/4, break up this group in step 3 by multiplying by whatever is on the bottom.
