When you get to the co-ordinate geometry bit of C1, suddenly the equation of a line changes from $y = mx + c$ to $(y-y_0) = m(x - x_0)$.

It doesn’t really change - it’s just another way of writing the same thing; especially when you’re dealing with a fractional gradient, this is a much nicer form to work with. (Especially especially when you need to give an answer in the form $ax + by + c = 0$, because you can simply multiply both sides by the bottom of the fraction and deal with whole numbers from the get-go.)

But what’s the difference between $x$ and $x_0$?

Oh, it’s simple. $(x,y)$ is a point on the line. And $(x_0, y_0)$ is a specific point on the line. Wait, that didn’t clear anything up.

OK, so you know how in $y = mx + c$, when you ended up with an equation, there was always an $x$ and a $y$ in it ((unless you were dealing with a horizontal or vertical line, smart-arse)) ? That’s true of any curve, which means it’s true for a straight line. Your equation is always going to have an $x$ and a $y$ in it.

$(x_0, y_0)$ is any point you know is on the line. That’s likely to be a point you’re given, or a point you’ve worked out. You can normally replace $x_0$ and $y_0$ with numbers.

And that’s it! $(x,y)$ is the general point on the line - you check if a point is on a line by checking $x$ and $y$ satisfy the equation; $(x_0, y_0)$ is a specific point you know to be on the line, and you should replace the co-ordinates with the numbers you know.