Lesson 1: Graphing Linear Equations

Property

An equation of the form $y = ax + b$, where $a$ and $b$ are constants, is called a linear equation because its graph is a straight line. We can graph a linear equation by evaluating the expression $ax + b$ at several values of $x$ and then plotting the resulting points.

Examples

• A sapling is 5 inches tall and grows 2 inches each week. The height $H$ after $w$ weeks is given by the equation $H = 5 + 2w$. After 3 weeks, the height is $H = 5 + 2(3) = 11$ inches.

• You are 200 miles from home and driving away at 60 miles per hour. Your distance $D$ from home after $h$ hours is $D = 200 + 60h$. After 2 hours, the distance is $D = 200 + 60(2) = 320$ miles.

Property

Graph a linear equation by plotting points.
Step 1. Find three points whose coordinates are solutions to the equation. Organize them in a table.
Step 2. Plot the points in a rectangular coordinate system. Check that the points line up. If they do not, carefully check your work.
Step 3. Draw the line through the three points. Extend the line to fill the grid and put arrows on both ends of the line.

Examples

• To graph $y = x + 3$, find three solution points. If $x=0, y=3$, giving $(0, 3)$. If $x=1, y=4$, giving $(1, 4)$. If $x=-1, y=2$, giving $(-1, 2)$. Plot these points and draw the line through them.
• To graph $y = 2x - 1$, find three points. If $x=0, y=-1$, so $(0, -1)$. If $x=2, y=3$, so $(2, 3)$. If $x=-1, y=-3$, so $(-1, -3)$. Plot these and connect them.
• To graph $y = \frac{1}{3}x + 2$, choose multiples of 3 for $x$ to avoid fractions. If $x=0, y=2$, so $(0, 2)$. If $x=3, y=3$, so $(3, 3)$. If $x=-3, y=1$, so $(-3, 1)$. Plot and connect.

Explanation

To draw a line, you only need a few of its points. Find three $(x, y)$ pairs that solve the equation, plot them on the coordinate plane, and connect them with a straight line. Using three points helps you catch any calculation mistakes.