Problem 59 - Introductory and Intermediate Algebra for College Students 4th

Question

Graph each equation. $$3 y=9$$

Step-by-Step Solution

Answer

The graph of the equation \(y=3\) is a straight horizontal line passing through the point (0,3).

Step 1: Simplify the Equation

Divide both sides of the equation by 3, resulting in the simplified form of the equation: \(y=3\).

Step 2: Identify Key Characteristics of the Graph

The graph is a horizontal line that cuts the y-axis at 3. This is because y is always 3, no matter what value x takes.

Step 3: Graph the Equation

Draw a horizontal line through the point (0,3) on the y-axis. This line represents the equation \(y=3\)

Key Concepts

Horizontal Lines

When we speak about horizontal lines in graphing linear equations, these lines are both intriguing and simple to understand. A horizontal line on a graph means that no matter the value of \( x \), the value of \( y \) remains constant. In other words, you are dealing with an equation where \( y \) appears alone on one side, such as \( y = 3 \).

The beauty of horizontal lines is in their consistency:

• They always run parallel to the x-axis.
• The slope of a horizontal line is 0, which signifies no vertical change as you move along the line.
• This stability makes them easy to graph!

For \( y = 3 \), the line will intersect the y-axis at the point (0,3) and extend indefinitely in both directions along the x-axis.

Simplifying Equations

Simplifying equations is a crucial step in the process of graphing linear equations. It's vital to reduce any complex equation into its simplest form to make the graphing process straightforward. Let's consider our example equation, \( 3y = 9 \).

To simplify, simply isolate \( y \) on one side of the equation. This typically involves operations such as:

• Dividing every term by a common factor
• Combining like terms, if needed

In this instance, divide both sides by 3 to achieve the simplified form \( y = 3 \).

Simplified equations not only make graphing easier, but they also assist in identifying key characteristics of a graph, such as slopes, intercepts, and overall behavior.

y-intercept

The y-intercept is a fundamental aspect of understanding and graphing linear equations. This is the specific point where the line crosses the y-axis. For horizontal lines like \( y = 3 \), determining the y-intercept is straightforward because the equation tells us directly where the line cuts the y-axis.

A few pointers about y-intercepts:

• For any equation of the form \( y = b \), the y-intercept is \( b \).
• It is represented on the graph by the point \( (0, b) \).
• This single point is vital for setting up your graph accurately.

In our previous example, since the equation simplifies to \( y = 3 \), the line crosses the y-axis at (0, 3), establishing this point as the y-intercept. Understanding y-intercepts helps to quickly ascertain a line’s starting position on the y-axis, making graphing much clearer.