Number Theory and Congruency I have the following problem:
$$2x+7=3 \pmod{17}$$
I know HOW to do this problem. It's as follows:
$$2x=3-7\\
x=-2\equiv 15\pmod{17}$$
But I have no idea WHY I'm doing that. I don't really even understand what the problem is asking, I'm just doing what the book says to do. Can someone explain what this problem is asking and what I'm finding? Thanks

Your original question is entirely equivalent (for $x,y\in \mathbb Z$) to: $$2x+7=3+17y$$ with the perspective that you want to know the value of $x$ and are not particularly interested in the value of $y$. Note that if you have found a solution $(x,y)$ and you set $x'=x+17k$ then:$$2x'+7=2x+34+7=3+17y+34=3+17(y+2)=3+17y'$$
where $y'=y+2$. So you can always find a solution with $0 \leq x \leq 16$.
Now we start on your solution, noting that $y$ must be even, so that we can put $y=2z$:$$2x+7=3+34z$$$$2x=-4+34z$$$$x=-2+17z$$
And we choose $z$ to give us the least nonnegative value of $x$, though this equation gives us the whole family of solutions.
Basically the notation of modular arithmetic enables the distinct roles of $x$ and $y$ to be formalised in such a way that these calculations can be done more efficiently and presented more economically, without attention to detail which turns out to be irrelevant.
