Slope-Intercept Form

This section specializes the point-slope form of Section 5 to the one particular point every non-vertical line is guaranteed to have: the point where it crosses the y-axis.

The y-intercept. A non-vertical line crosses the y-axis exactly once, at a point of the form $(0,c)$ for some real number $c$, called the y-intercept of the line.

Derivation. Apply the point-slope form of Section 5 with $(x_1,y_1)=(0,c)$:

Rearranging,

— the slope-intercept form. It displays the line's slope $m$ and y-intercept $c$ directly as the coefficients once the equation is solved for $y$: the coefficient of $x$ is the slope, and the constant term is the y-intercept.

Reading $m$ and $c$ off a general equation. Given any linear equation $Ax+By+C=0$ with $B\ne 0$, solving for $y$ gives $y = -\dfrac{A}{B}x - \dfrac{C}{B}$, so by direct comparison with $y=mx+c$: slope $m=-\dfrac{A}{B}$ and y-intercept $c=-\dfrac{C}{B}$. This is the fastest way to extract a line's slope from its general equation, without needing two explicit points on it.

Special case: line through the origin. When $c=0$, the slope-intercept form reduces to $y=mx$, the equation of every line through the origin with slope $m$.

Why $B\ne 0$ is required. A line parallel to the y-axis (Section 4, $x=a$) has $B=0$ in general form and no finite slope, so it has no y-intercept in this sense either (it crosses the y-axis only if $a=0$, and then coincides with the whole axis) — it is the one line shape that never has a slope-intercept equation. …