The formula
y − y₁ = m(x − x₁) → y = mx + b
Why it works
Two points determine a line, and the line has one equation written three different ways. Point-slope comes straight from the definition of slope; rearranging it gives slope-intercept; multiplying out and collecting gives the general form Ax + By + C = 0.
How to solve it by hand
- Work out the slope from the two points
- Substitute one point into y − y₁ = m(x − x₁)
- Expand and rearrange for y = mx + b
- Collect everything on one side for Ax + By + C = 0
What is worth knowing
Each form earns its place. Slope-intercept is best for graphing, because you can read the slope and the starting point straight off. Point-slope is best for constructing a line when you know a point and a direction, which is what you have most of the time. The general form is the one that survives the vertical case: x = 3 has no slope-intercept form at all, but it writes perfectly as 1x + 0y − 3 = 0. That robustness is why the general form is what computer graphics and linear algebra actually use.