Slope tile collision is deceptively difficult, depending on how many situations you want it to deal with. And indeed, there's very little info on the net about it. Particularly doing anything more than 45 degree tiles where you climb up with the corner of your collision box rather than the center/feet of the character.
One of these days I'll write an article on it... when I understand it better myself.
45 degree slopes, on the ground only (not ceiling) are probably doable for a relative newbie... I assume you already have solid tile collision working? What approach are you using, separate X and Y movement steps?
As for your line equation, for 45 degree tiles the slope is 1, so it cancels out... no multiplication is necessary. X and Y (within the tile) are always equal to eachother, either positively or negatively.
So assuming positive X is right, positive Y is down, and you're on a 45 degree slope tile where the bottom left is solid and top right is open... then to stand on the slope, figure your X position within the tile, and set Y within the tile equal to that. For the other direction (bottom right solid, top left open), set Y equal to tilesize - 1 - X. Notice the -1. That's one of the things that makes it so difficult. One direction is always off by 1 pixel from the other.
Another problem is that when walking up a slope onto flat ground, with your center point resting on the slope, then the bottom corner of your collision box will ruin into the "wall" of the flat ground tile before you get to the top of the slope.
Draw lots of diagrams and study the situations that come up when you're in the air, walking on the ground, walking up a slope, down a slope, whether there's a flat ground tile at the top/bottom of the slope, or a wall... lots of situations, and in my experience, there's no simple, elegant solution.