In the first formula, you start with two points, whose coordinates are (x1,y1) and (x2,y2). The difference y2 − y1, which is called the rise, is how much you move up as you go from the first point to the second; if you actually move down, then this is negative. The difference x2 − x1, which is called the run, is how much you move to the right; if you actually move to the left, then this is negative. The rise and run vary depending on which points you look at, but if you divide the rise by the run, then you'll get the same result no matter which two points you choose, as long as you pick two distinct points on the same line.
This number, which is usually denoted by m, is the slope, and it describes the directions in which you can travel along the line. Lines with positive slope run up–right and down–left; lines with negative slope run down–right and up–left. Lines whose slope has a large absolute value are steep; lines whose slope has a small absolute value are shallow. Horizontal lines have a slope of exactly zero. (For vertical lines, see below.)
The second formula tells you two things: If you know the slope m of a line and you know its y-intercept (0,b), then by putting those numbers (m and b) into the second formula, you get an equation (in x and y) for the line. Even if you don't know the y-intercept, as long as you know the slope, then you can use the coordinates of any point to get an equation for b; solve this, and now you can write down the equation for the line.
In the other direction, if you have an equation for the line, then solve it for y. By comparing this with the second formula above, you now know what the slope and y-intercept are. Once you have the slope and any point, it's easy to draw a graph.
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