Business Mathematics and Statistics · Ch 3 — Analytical Geometry (Locus, Straight Lines, Pair of Straight Lines, Circles, Conics)

Equation of a Straight Line — Standard Forms

A straight line can be written in several equivalent algebraic forms; which one to use depends on what information the problem gives — a point and a slope, two points, or the two axis-intercepts. Every form can always be rearranged into the single general form $ax + by + c = 0$ .

Slope (point-slope) form — given one point $(x_1, y_1)$ on the line and its slope $m$ :

y - y_1 = m(x - x_1)

Two-point form — given two points $(x_1,y_1)$ and $(x_2,y_2)$ on the line, the slope is $m = \dfrac{y_2-y_1}{x_2-x_1}$ (for $x_1 \ne x_2$ ), and the equation is:

y - y_1 = \dfrac{y_2-y_1}{x_2-x_1}(x - x_1)

Intercept form — given the $x$ -intercept $a$ and the $y$ -intercept $b$ (the points where the line crosses each axis):

\dfrac{x}{a} + \dfrac{y}{b} = 1

Figure 1 — Line 4x − y − 11 = 0 through (2, −3) with a slope triangle (rise 4, run 1) illustrating slope 4

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Definition 1Slope (Gradient) of a Line

For a line through $(x_1,y_1)$ and $(x_2,y_2)$ , the slope is $m = \dfrac{y_2-y_1}{x_2-x_1}$ — the rise per unit run. A positive slope rises left to right, a negative slope falls, a slope of $0$ is horizontal, and …

Definition 2Point-Slope and Two-Point Forms

Point-slope form $y-y_1=m(x-x_1)$ is used when a point and the slope are known; two-point form $y-y_1=\dfrac{y_2-y_1}{x_2-x_1}(x-x_1)$ is used when two points on the line are known instead — it is really point-slo …

Definition 3Intercept Form and General Form

Intercept form $\dfrac{x}{a}+\dfrac{y}{b}=1$ is convenient when the axis crossings are known directly. The general (standard) form $ax+by+c=0$ is the form every other form reduces to, and is the one most useful …

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