Writing the Equation of a Line from Points or a Slope

A line is just a constant rate of change. If you know how steep it is and you know one place it passes through, there is exactly one line that fits — nothing else is allowed to wiggle. That is why a single point plus a slope locks a line in place as firmly as two points do: two points secretly give you both pieces anyway, since you can always compute the slope between them first.

Slope stays constant along a line, which is the whole idea behind the rise-over-run triangle. Draw a triangle between any two points on the line, with a vertical leg (rise) and a horizontal leg (run). Slide that triangle anywhere else on the line, or stretch it wider, and the legs grow or shrink together, keeping rise/run the same number. That constancy is exactly what slope measures.

Point-slope form, y - y1 = m(x - x1), captures a point (x1, y1) and a slope m directly, with no algebra needed first. Once it is written, distribute and solve for y to land on slope-intercept form, y = mx + b, which exposes the y-intercept. This skill feeds directly into systems of equations, where you compare two lines' slopes and intercepts, and into interpreting graphs, where slope and intercept describe a real rate and starting value.