Problem 66 - Elementary Algebra Within Reach

Question

In Exercises \(63-66\), sketch the graph of the equation. $$ x=6 $$

Step-by-Step Solution

Answer

The graph of the equation \(x=6\) is a vertical line crossing the x-axis at x=6.

Step 1: Identify the type of line

The given equation is \(x = 6\). This equation represents a vertical line that passes through x=6 on the x-axis in the Cartesian coordinate system. This equation means no matter what y is, x will always be 6.

Step 2: Plot the line

To sketch this graph, draw a vertical line that passes through the point (6,0) on the x-axis. The line will go straight up and down, and this line represents all points where x=6. Because there is no y in the equation, this line extends to all values of y.

Key Concepts

Cartesian coordinate system

The Cartesian coordinate system is essential for graphing equations and understanding geometry. It is made up of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). These axes intersect at a point called the origin, denoted as (0,0). Each point on the plane is defined by a pair of numbers, known as coordinates. These coordinates specify the location of the point based on its distance from the axes.
The first number, the x-coordinate, indicates the position along the x-axis, while the second number, the y-coordinate, provides the position along the y-axis.

• For example, the point (6,0) is 6 units to the right of the origin along the x-axis and lies on the x-axis itself, with no vertical displacement.

Understanding this system is crucial for graphing and interpreting different types of lines, including vertical lines like the one in the given equation.

Vertical line equation

A vertical line in the Cartesian coordinate system is represented by an equation of the form \(x = a\), where \(a\) is any real number. This equation indicates that for every point on the line, the x-coordinate remains constant at \(a\), regardless of the value of \(y\).
For example, with the equation \(x = 6\), every point on this vertical line has an x-coordinate of 6. So, some of the points could be (6, 1), (6, -2), or (6, 5), extending infinitely in both the positive and negative directions on the y-axis.

• A vertical line does not have a slope because the rise over run formula leads to a division by zero situation, which is undefined.

This distinct characteristic makes vertical line equations straightforward to graph, as they require knowing only the constant x-value.

Plotting graphs

Plotting graphs, especially vertical lines, can be simple once you understand the basics. Begin by identifying the type of line you need to draw from the equation. In the case of a vertical line such as \(x = 6\), the instructions come directly from the equation.
Here's a quick guide on how to sketch this graph:

• First, locate the x-coordinate on the x-axis, which is 6 in this case.
• Next, draw a line parallel to the y-axis, passing through this x value.
• The line is vertical because its direction is straight up and down, covering any y-value possible, meaning it runs through points like (6, -1), (6, 0), and (6, 2).

This procedure is useful for visualizing data and understanding the relationship represented by different equations. Remember, plotting involves accurately transcribing the equation's information onto the coordinate plane for precise interpretation.