Forms of the Equation of a Line

The slope-intercept form of (the equation of) a line is: $y\=m x\+b$. The slope of the line is $m$. The y-intercept of the line is b, as can be seen by substituting $x\=0$ into the equation.

The point-slope form of (the equation of) a line is: $y-y_1\=m \left(x-x_1\right)$. The slope of the line is $m$. The point $\left(x_1\,y_1\right)$ is on the line, as can be seen by substituting $x\=x_1\, y\=y_1$ into the equation and observing that you get $0\=0$.

The two-point form of (the equation of) a line is: $y-y_1\=\frac{y_2-y_1}{x_2-x_1}\left(x-x_1\right)$. The points $\left(x_1\,y_1\right)$ and $\left(x_2\,y_2\right)$ are on the line, as can be seen by substituting them into the equation and observing that the equality holds. It may be easier to remember this form when rewritten in the symmetrical form $\frac{y-y_1}{y_2-y_1}\=\frac{x-x_1}{x_2-x_1}$.

The general (or implicit) form of (the equation of) a line is: $a x\+b y\=c$, or, alternatively, $a x\+b y\+c\=0$. (Note that the c's in these two versions of this form are not the same - one is the negative of the other.) By convention, the coefficient of the x term, a, is taken to be positive, but this is a convenience, not a rule.

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