Mathematics · Ch 5 — Straight Line

Point-Slope Form

5.3.1

Point-Slope Form

5.3.1 Point-Slope Form

Goal. Find the equation of the line with slope $m$ passing through a known point $A(x_1,y_1)$ .

Proof. Let $L$ be the line through $A(x_1,y_1)$ with slope $m$ , and let $P(x,y)$ be any other point on $L$ . The slope of $L$ , computed between $A$ and $P$ , is $\dfrac{y-y_1}{x-x_1}$ ; but this must equal the given slope $m$ . So

\frac{y-y_1}{x-x_1} = m \implies (y-y_1) = m(x-x_1).

This is the point-slope form. (In particular, if the line passes through the origin with slope $m$ , its equation is simply $y = mx$ .) …

Figure 5.3.1-Fig5.9Fig. 5.9

What this figure shows. Diagram showing the fixed point A(x1,y1), a general point P(x,y) on the line, and the line of slope m used to derive the point-slope form. This figure gives the reader a concrete visual reference for the geometric configuration described in the surrounding text, tying the abstract statement to …

Misc 5.3.1-Ex1Worked example — point-slope form

Worked out. Finds the equation of the line through A(2,1) with slope −3 by substituting directly into the point-slope formula and simplifying. Working through this worked example after reading the theory above helps consolidate the method before attempting the exercise questions that follow it in the tex …

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