Question

How do you solve `13=5-13+3a`?

Answer 1

There is only one correct answer, but there are many ways to get to that answer.
(There is only one London, England, but many ways to get there.)

Here's one:
Look for things the can be combined on the left of the equal sign. No, there's only a 13 there.
Look for things that can be combined on the right of the equal sign. Yes, I can do 5-13. So we write:

`13=5-13+3a` is true if and only if:

`13=-8+3a`.

Often the words "is true if and only if" clutter up our solution, so we leave them out. I'll leave them out from here on.

Now, I want an `a` that makes it true that `13=-8+3a`.
Our goal is to get `a` alone on one side `=` some number on the other side. `a= "some number"`
A lot of people like to put the unknown (`a`) on the left.
There are two ways of doing this. I think the simpler is to use the fact that `"this" = "that"` exactly when `"that" = "this"`
So now we want:

`-8+3a=13`

Look at the left side. Order of operations: if I picked a number for `a` what arithmetic would I do in what order? First I'd multiply by 3. The I'd add `-8` (or subtract `8`).

To get `a` alone, we'll undo those operations. First we'll add `+8`, then we'll divide by `3`.

The whole process, without words explaining it looks like this:

`13=5-13+3a`
`13=-8+3a`
`-8+3a=13`
.
`color(red)(3a=13+8)`
.
`3a=21`
`(3a)/3 = 21/3`
`a=7`

The solution to the last equation is evident. The solution is `7`.
The solution to the first equation is also `7`

I left some blank lines. For many people it becomes tiresome to include the steps below in red: (The middle 2 lines.)
`-8+3a=13`
`color(red)((-8+3a)+8=13+8)`
`color(red)(-8+3a+8=21)`
`3a=21`

Here's a slightly different looking solution

`13=5-13+3a`

`5-13+3a=13`

`3a=13-5+13`

`3a=21`

`a=21/3=7`

Again, it is obvious that the solution to the last equation is `7`.

The solution to the first equation is also `7`