Writing the Equation of a Line

A line is infinite, but it has no freedom once you know two things about it: how steep it is, and where it sits. Slope tells you the direction. One point tells you the location. Together they rule out every other line with that same slope and every other line through that same point, leaving exactly one. That's why point-slope form, y - y1 = m(x - x1), works: plug in the point (x1, y1) and the slope m, and you've written down the full set of (x, y) pairs that satisfy both conditions at once.

When you're handed two points instead of a point and a slope, do the slope step first: m = (y2 - y1) / (x2 - x1). Once you have m, treat either point as your (x1, y1) and drop both into point-slope form. From there, distribute and isolate y to land in slope-intercept form, y = mx + b, if that's what's wanted. The order matters: you cannot use y = mx + b to solve for b until m is already known, because b is only the leftover constant after slope and one point have done their work. Skipping straight to y = mx + b with an unknown m is the shortcut that goes nowhere.

Vertical lines break this pattern entirely. Every point on a vertical line shares the same x-coordinate, so there's no rise-over-run slope to compute — the run is zero, and division by zero doesn't exist. The equation is just x = a, a constant, with no y in it at all. This sits right next to horizontal lines (y = b, slope zero) as the two special cases worth recognizing on sight, before you move on to midpoint and distance formulas that also lean on coordinate pairs.