How to find the inverse/unit for mod? So for this problem, its asking to find the inverses of [4], [5], and [7]. I'm trying to skim through my textbook but I can't find an example.

Like for this problem, I don't understand why the Units are 1 and 2, the textbook I have is garbage.
Examples:

For a) you can find the inverses like this.

For example when we want to find the inverse  of $[4]$ in $\mathbb{Z}/13\mathbb{Z}$.
Since $13$ is a prime we know, that $\mathbb{Z}/13\mathbb{Z}$ is a field.

To find the inverse $[4^{-1}]$ we have to solve $[4][x]=[4x]=[1]$.
We can do this by brute force and just go all the way. There are just 13 possibilities for $x$.

One sees pretty fast, that $[4\cdot 10]=[40]=[1+13\cdot 3]=[1]$. Hence we have $[4^{-1}]=[10]$

This is a basic approach.
An other is to use the euclidean algorithm.

Now go find the other inverses!

For b) we can use the results in a).
The equations can be solved as always:

$[4]X=[7]$.

We want to isolate $X$. For that we multiply with $[4^{-1}]$, and get:

$[10][4]X=[10][7]$

$X=[70]=[5]$
Your question（２） is to solve equations 
in mod13,
[1]4x≡７mod13
[2]5x≡１２mod13
[3]7x≡８mod13.

I will explain using [1]4x≡7mod13.

'To solve this equation' means what?

That is, 'To deformation the former equation[4x≡7mod13] to [x≡●mod13] and find  ●'.

What can we do to do such a thing?

Of couse, if only we could change 4 to 1 !

How can we do such a magical thing ?
No, No..., this is not a magical thing...

You can simply multiply on both side of equation from left side , 10 !
This number,10 is the number what we needed! And such a number　（satisfying ●ｘ≡１mod13
）is called inverse　of 13. 

Generally, in mod p[p is prime], it is guaranteed that we can find the inverse in ｛0,1,2,...,p-1｝ uniquely .

Once you get　the inverse, you can solve equation in mod13 very easily like below.
 4x≡7mod13
4×10x≡7×10mod13
x≡70mod13＝5mod13 □

So,　if you say, problem（１）　is a very kind Hint for us to solve the equation problem（２）！

Then, I guess you understood you can solve the equation modp[p;prime] once you solve the inverse. 

Then , how can we find The way to find the inverse　in concrete method?

The answer is quite simple ! The candidate of inverse only in 0,1,2,3,・・・,12 as I mentioned in the previous sentense !
So you just need to try and error and check these 13 number and find 10.

If you want to find the inverse more efficiently, you can omit the process and find the inverse a like below.

4a≡1mod13
12a≡3mod13
-a≡3mod13
a≡10mod13