Standard Forms of the Equation of a Straight Line

A straight line is fixed by knowing enough about it — a point and a direction, or two points, or its intercepts. Each kind of information leads to a convenient standard form. All of them describe the same object; choose whichever matches the data given in a problem.

Slope–intercept form. If the line has slope $m$ and cuts the y-axis at $(0, c)$ (so $c$ is the y-intercept):

Point–slope form. If the line has slope $m$ and passes through a known point $(x_1, y_1)$:

Two–point form. If the line passes through two known points $(x_1, y_1)$ and $(x_2, y_2)$:

This is just the point–slope form with the slope computed from the two points.

Intercept form. If the line cuts the x-axis at $(a, 0)$ and the y-axis at $(0, b)$, where $a$ is the x-intercept and $b$ the y-intercept (both non-zero):

Normal (perpendicular) form. If the perpendicular dropped from the origin onto the line has length $p$ ($p > 0$) and makes an angle $\alpha$ with the positive x-axis: