Section 4.1: 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

The graph of a linear equation in the form $y = mx + b$ is a line.

• Every point on the line is a solution of the equation.
• Every solution of this equation is a point on this line.

Property

A horizontal line is the graph of an equation of the form $y = b$. The line passes through the $y$-axis at $(0, b)$.
In this type of equation, the value of $y$ is always equal to $b$, no matter the value of $x$.

Examples

• The graph of the equation $y = 4$ is a horizontal line where every point has a y-coordinate of 4, such as $(0, 4)$, $(2, 4)$, and $(-1, 4)$.
• The equation $y = -1$ represents a horizontal line that passes through the y-axis at the point $(0, -1)$.
• A horizontal line that passes through the point $(5, -6)$ has the equation $y = -6$ because the y-coordinate must always be -6.

Explanation

When an equation only has a $y$ variable, like $y = 3$, it means $y$ is always fixed at that number. No matter how far left or right you move along the x-axis, the result is a perfectly flat, horizontal line.