The Timpson Lyceum · Intermediate Algebra (2023)

Lecture 4 (Intermediate Algebra) Graphs and Equations of Lines

Overview

This lecture covers the coordinate plane and the equations of lines — the heart of introductory analytic geometry. You will learn how to plot points, graph a line from its equation, and calculate the slope between two points. From there we build the three standard ways to write a line: slope-intercept form, point-slope form, and the equations of horizontal and vertical lines. We close with the relationships between slopes of parallel and perpendicular lines. These tools let you move freely back and forth between a line's picture and its equation.

Key Concepts

The coordinate plane. Points are plotted on the $xy$ -plane as ordered pairs $(x, y)$ . The first coordinate $x$ tells you how far to move horizontally from the $y$ -axis (positive = right, negative = left); the second coordinate $y$ tells you how far to move vertically from the $x$ -axis (positive = up, negative = down). A generic point with unknown coordinates is written $(x_1, y_1)$ .

Graphing a line from an equation. An equation like $y = 2x$ is satisfied by infinitely many $(x, y)$ pairs. Plotting several of those points reveals that they all lie on one straight line — the line is the set of all points satisfying the equation. To draw a line you only need two points, since exactly one line passes through any two distinct points. In practice, build a small table of $x$ -values, compute the matching $y$ -values, plot the points, and draw the line through them.

Slope. Slope measures steepness as rise over run. Given two points $(x_1, y_1)$ and $(x_2, y_2)$ on a line, the slope (denoted $m$ ) is the change in height divided by the change in horizontal distance: $m = \frac{y_2 - y_1}{x_2 - x_1}$ It does not matter which point you label "1" and which you label "2," as long as you are consistent in the numerator and denominator.

Slope-intercept form. $y = mx + b$ , where $m$ is the slope and $b$ is the $y$ -intercept (the $y$ -value where the line crosses the $y$ -axis, i.e. the point $(0, b)$ ). This is the most common way to write a line because it displays both the slope and the intercept at a glance.

Point-slope form. Starting from $m = \dfrac{y - y_1}{x - x_1}$ and clearing the denominator gives $y - y_1 = m(x - x_1)$ Use this when you know one point $(x_1, y_1)$ on the line and the slope $m$ . (Substituting the point $(0, b)$ into point-slope form recovers slope-intercept form.)

Horizontal and vertical lines.

$y = a$ is a horizontal line at height $a$ . Its slope is $0$ (the rise is always $0$ ).
$x = a$ is a vertical line through $a$ on the $x$ -axis. Its slope is undefined, because the run is $0$ and you cannot divide by zero.

Parallel lines. Two lines are parallel when they never intersect, which happens exactly when they have equal slopes (but different $y$ -intercepts).

Perpendicular lines. Two lines meet at right angles when their slopes are negative reciprocals of each other. If one line has slope $m$ , the perpendicular line has slope $-\dfrac{1}{m}$ (flip the fraction and change the sign). For example, slope $2$ is perpendicular to slope $-\tfrac{1}{2}$ .

Worked Example

Find the equation of the line through $(6, 9)$ that is perpendicular to $3x - 5y = 2$ , written in slope-intercept form.

First find the slope of the given line by solving for $y$ , then take the negative reciprocal to get the perpendicular slope:

\begin{aligned} 3x - 5y &= 2 \\ -5y &= -3x + 2 \\ y &= \tfrac{3}{5}x - \tfrac{2}{5} \quad\Rightarrow\quad m_{\text{given}} = \tfrac{3}{5} \\ m_{\perp} &= -\tfrac{5}{3} \end{aligned}

Now use point-slope form with the point $(6, 9)$ and slope $-\tfrac{5}{3}$ , then solve for $y$ :

\begin{aligned} y - y_1 &= m(x - x_1) \\ y - 9 &= -\tfrac{5}{3}(x - 6) \\ y - 9 &= -\tfrac{5}{3}x + 10 \\ y &= -\tfrac{5}{3}x + 19 \end{aligned}

Summary

Plot points as $(x, y)$ : first coordinate moves left/right, second moves up/down.
A line is the set of all points satisfying its equation; two points are enough to graph it.
Slope is $m = \dfrac{y_2 - y_1}{x_2 - x_1}$ — rise over run between any two points on the line.
Slope-intercept form $y = mx + b$ shows slope and $y$ -intercept directly; point-slope form $y - y_1 = m(x - x_1)$ is built from one point and a slope.
Horizontal lines $y = a$ have slope $0$ ; vertical lines $x = a$ have undefined slope.
Parallel lines share the same slope; perpendicular lines have negative-reciprocal slopes.