Summary

This chapter developed the algebra of the straight line from its two most basic descriptors — a slope and a point (or two points, or an intercept) — building every named form from the single point-slope idea.

• The slope $m=\tan\theta$ of a line measures its steepness via its inclination $\theta$; between two known points it is computed as $m=\dfrac{y_2-y_1}{x_2-x_1}$. Three points are collinear exactly when the slope between each pair agrees.
• The angle between two lines of slopes $m_1,m_2$ is $\theta=\tan^{-1}\left|\dfrac{m_1-m_2}{1+m_1m_2}\right|$, giving the corollaries $L_1\parallel L_2 \iff m_1=m_2$ and $L_1\perp L_2 \iff m_1m_2=-1$.
• A line parallel to an axis has the simplest possible equation: $y=b$ (parallel to the x-axis) or $x=a$ (parallel to the y-axis, where the slope is undefined).
• Every other form follows from the point-slope form $y-y_1=m(x-x_1)$: setting the point to the y-intercept $(0,c)$ gives the slope-intercept form $y=mx+c$; substituting a slope found from two given points gives the two-point form $\dfrac{y-y_1}{y_2-y_1}=\dfrac{x-x_1}{x_2-x_1}$; and specializing the two-point form to the axis-intercepts $(a,0),(0,b)$ gives the intercept form $\dfrac{x}{a}+\dfrac{y}{b}=1$. …