Lesson 3: Graphs of Linear Equations

New Concept

A linear equation, written as $y = ax + b$, forms a straight line when graphed. We'll explore how to plot these lines on a Cartesian coordinate system and use them to visually solve problems and analyze relationships.

What’s next

Now, let's move to interactive examples where you'll build these graphs by creating tables of values and plotting points on the coordinate plane.

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

To make a graph that includes negative values, we construct a Cartesian coordinate system. We draw two perpendicular number lines for the horizontal and vertical axes. The horizontal axis is called the $x$-axis and the vertical axis is the $y$-axis. The two axes divide the plane into four quadrants. The axes intersect at the origin, which has coordinates $(0, 0)$.

Examples

• To plot the point $(4, 3)$, we start at the origin, move 4 units to the right along the x-axis, and then 3 units up. This point is in the first quadrant where both coordinates are positive.

• The point $(-5, 2)$ is located 5 units to the left of the y-axis and 2 units above the x-axis. It lies in the second quadrant.

Property

The graph of an equation is a picture of the solutions of the equation. Each point on the graph represents a solution. If a point lies on the graph, its coordinates make the equation true. When you graph a linear equation, you should extend the line far enough in both directions so that it will cross both the $x$-axis and the $y$-axis.

Examples

• For the equation $y = 3x - 1$, the point $(2, 5)$ is a solution because when we substitute $x=2$, we get $y = 3(2) - 1 = 5$. Therefore, the point $(2, 5)$ lies on the graph of the line.

• To solve $3x - 1 = 8$ using the graph of $y = 3x - 1$, we find the point on the line where the y-coordinate is 8. The corresponding x-coordinate for that point is 3, so $x=3$ is the solution.

Property

If the coefficient of $x$ is a fraction, we can make our work easier by choosing multiples of the denominator for the $x$-values. That way we won't have to work with fractions to find the $y$-values.

Examples

• To graph $y = \frac{1}{4}x + 2$, choose x-values that are multiples of 4, such as -4, 0, and 4. If $x=4$, then $y = \frac{1}{4}(4) + 2 = 1 + 2 = 3$. The point is $(4, 3)$.

• For the equation $y = -1 + \frac{2}{5}x$, we can choose x-values like -5, 0, and 10. If $x=10$, then $y = -1 + \frac{2}{5}(10) = -1 + 4 = 3$. The point is $(10, 3)$.