Mathematics · Ch 5 — Straight Line

Double-Intercept Form

5.3.4

Double-Intercept Form

5.3.4 Double-Intercept Form

Goal. Find the equation of the line making non-zero intercepts $a$ (on the X-axis) and $b$ (on the Y-axis).

Proof. The line meets the X-axis at $A(a,0)$ and the Y-axis at $B(0,b)$ . For any other point $P(x,y)$ on the line, the slope of $AP$ equals the slope of $AB$ :

\frac{y-0}{x-a} = \frac{b-0}{0-a} = -\frac{b}{a}.

Cross-multiplying and rearranging,

\frac{y}{x-a} = -\frac{b}{a} \implies \frac{x}{a} + \frac{y}{b} = 1 \qquad (a,b \ne 0).

This is the double-intercept form. …

Figure 5.3.4-Fig5.12Fig. 5.12

What this figure shows. Diagram showing the line crossing the axes at A(a,0) and B(0,b), used to derive the double-intercept form. This figure gives the reader a concrete visual reference for the geometric configuration described in the surrounding text, tying the abstract statement to a pic …

Misc 5.3.4-Ex1Worked example — double-intercept form

Worked out. Finds the equation of the line making intercepts 3 and 4 on the coordinate axes by substituting directly into x/a + y/b = 1. Working through this worked example after reading the theory above helps consolidate the method before attempting the exercise questions that follow it in the textbook …

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