I have a question on the calculations in linear congruences if somebody could please clear up for me?

Say for example, the linear congruence:

We check that there are solutions: & incongruent solutions.

Then simplify to:

Then check if they are relatively prime s.t. there exists a multiplicative inverse, , such that :

, such that .

The we use the Euclidean Algorithm to find :

Then going back to the original question:

This was an example in class, but I do believe the last line should say:

My question is:

1. - How do we get the multiplicative inverse from the Euclidean Algorithm?

- Specifically, how does one get from the Euclidean Algorithm?

- Which 3 are we looking at in the algorithm?

2. - How does the original question of translate to ?

- Why is suddenly changed to 7?

Other than that, I think I understand all of the other steps. Just couple of small hiccups in the theory I don't understand. Any help would be much appreciated. Thanks.