number theory - Using the euclidean algorithm to find the inverse of $50 mod 3$

To solve a congruency like

$$50x \equiv 17 \mod3$$

You need to find the inverse of

$$50x \mod 3$$

For this, you have to write $1$ as a linear combination of $50$ and $3$:

$$1 = 50k_1+3k_2$$

Good, now how do I find $k_1$ and $k_2$... Well, the paper says that to do so, I must

Use the Euclidean algorithm. Hence:

$$1 = 3 – 2*1\\ 1 = 3 - (50 – 3*16)*1 = 3(1+16) + 50*(-1) = 3*17 -1*50$$

Now, now. I know how to do that algorithm to find the greatest common divisor. I also know that such value for $50$ and $3$ should be $1$ so that they can have an inverse.

But that's it. I don't know how to "use" the Euclidean algorithm to "solve" the congruency. I know the steps to the algorithm and yet do not understand what did they do in the quote above. Could you clarify?