DE-BOOK Equation of a Line in Point-Slope Form

Skip to main content

\(\newcommand\DLGray{\color{Gray}} \newcommand\DLO{\color{BurntOrange}} \newcommand\DLRa{\color{WildStrawberry}} \newcommand\DLGa{\color{Green}} \newcommand\DLGb{\color{PineGreen}} \newcommand\DLBa{\color{RoyalBlue}} \newcommand\DLBb{\color{Cerulean}} \newcommand\ds{\displaystyle} \newcommand\ddx{\frac{d}{dx}} \newcommand\os{\overset} \newcommand\us{\underset} \newcommand\ob{\overbrace} \newcommand\obt{\overbracket} \newcommand\ub{\underbrace} \newcommand\ubt{\underbracket} \newcommand\ul{\underline} \newcommand\laplacesym{\mathscr{L}} \newcommand\lap[1]{\laplacesym\left\{#1\right\}} \newcommand\ilap[1]{\laplacesym^{-1}\left\{#1\right\}} \newcommand\tikznode[3][] {\tikz[remember picture,baseline=(#2.base)] \node[minimum size=0pt,inner sep=0pt,#1](#2){#3}; } \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \definecolor{fillinmathshade}{gray}{0.9} \newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}} \newcommand{\sfrac}[2]{{#1}/{#2}} \)

Section Equation of a Line in Point-Slope Form

You’ve likely encountered the familiar slope-intercept form \(y = mx + b\) for the equation of a line. While this form is helpful, there’s another version that plays a more direct role in differential equations: the point-slope form. It’s especially useful when we know a point on the line and the slope, which is exactly the situation we encounter in slope fields and Euler’s method.

Recall the formula for the slope between two points, \((x_1, y_1)\) and \((x, y)\text{:}\)

\begin{equation*} m = \frac{y - y_1}{x - x_1} \end{equation*}

Rearranging this gives the point-slope form of a line:

\begin{equation*} y - y_1 = m(x - x_1) \end{equation*}

This form is powerful when modeling rates of change. For instance, when approximating solutions to differential equations numerically, each step follows the slope of a tangent line—exactly what this formula captures. You can also solve for \(y\) if needed:

\begin{equation*} y = m(x - x_1) + y_1 \end{equation*}

🌌 Example 307. Writing a Line from Two Points.

Given the points \((3, 7)\) and \((-1, 2)\text{:}\)

Write the equation of the line in point-slope form.

🔗
Solve the equation for \(y\text{.}\)

🔗

Solution.

First, find the slope:

\begin{align*} m \amp = \frac{2 - 7}{-1 - 3} = \frac{-5}{-4} = \frac{5}{4} \end{align*}

Now plug into the point-slope form using either point:

\begin{align*} y - 7 \amp = \frac{5}{4}(x - 3) \quad \text{or} \quad y - 2 = \frac{5}{4}(x + 1) \end{align*}

Solving for \(y\) gives:

\begin{align*} y \amp = \frac{5}{4}(x - 3) + 7 \quad \text{or} \quad y = \frac{5}{4}(x + 1) + 2 \end{align*}

Exercises Practice Writing Equations of Lines

For each pair of points below:

Write the equation of the line in point-slope form.

🔗
Solve the equation for \(y\text{.}\)

🔗

1.

\((6, -1)\) and \((2, -9)\)

2.

\((5, 0)\) and \((1, 2)\)

3.

\((-3, -7)\) and \((0, 4)\)

You have attempted of activities on this page.