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Chapter 4: Problem 111

In the following exercises, graph each equation. \(x=\frac{7}{3}\)

Short Answer

Expert verified
Draw a vertical line through \(x = \frac{7}{3}\).

Step by step solution

01

Identify the Type of Equation

Recognize that the equation given is in the form of a vertical line, which is represented by the equation \(x = k\), where \(k\) is a constant.
02

Determine the Value of x

The given equation is \(x = \frac{7}{3}\). This means that for any value of y, x will always be equal to \(\frac{7}{3}\).
03

Plot the Line on the Graph

To graph the equation, mark the point \(\frac{7}{3}\) on the x-axis. Since the equation represents a vertical line, draw a straight vertical line through this point extending infinitely in both the positive and negative y-directions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vertical Line Equation
Vertical lines are a unique type of equation in graphing.
Unlike other equations that express a relationship between x and y, a vertical line equation simply states that x always equals a specific constant.
The general form for a vertical line equation is written as:
\(x = k\)
Here, \(k\) is this constant. In our case, we have \(x = \frac{7}{3}\). This means no matter what value y takes on the graph, x will stay constant at \(\frac{7}{3}\).
It is crucial to remember that vertical lines have distinctly unique properties:
  • They are parallel to the y-axis.
  • They do not have a slope, making their slope undefined.
  • They intersect the x-axis at one point.
Plotting Points
Plotting points on a graph is a fundamental skill in understanding and visualizing equations.
A point in a two-dimensional space is represented as an ordered pair (x, y). Each part of the pair tells you where to plot the point on a coordinate grid:
  • The first number represents the x-coordinate, which shows how far left or right the point is.
  • The second number is the y-coordinate, which shows how far up or down the point is.
Let鈥檚 break it down with our vertical line equation \(x = \frac{7}{3}\). To plot this:
  • Go to \(x = \frac{7}{3}\) on the x-axis.
  • It doesn't matter what y value you pick because x is constant. Mark several points along x = \(\frac{7}{3}\), such as (\frac{7}{3}},1), (\frac{7}{3}, -1), etc.
Once these points are plotted, connecting them forms the vertical line.
Coordinate System
The coordinate system, also known as the Cartesian plane, is a two-dimensional surface used for graphing points, lines, and curves.
This system has two axes:
The x-axis runs horizontally and the y-axis runs vertically. They intersect at the origin, labeled (0, 0).
Here are some key features of the coordinate system:
  • The x-axis divides the plane into upper and lower halves.
  • The y-axis divides the plane into left and right halves.
  • Each point on the plane is identified by its coordinates (x, y).
To graph our equation \(x = \frac{7}{3}\),
  • First, locate \(\frac{7}{3}\) on the x-axis.
  • Then draw a vertical line that passes through this x-coordinate, moving infinitely up and down.
This helps visualize the equation effectively on the Cartesian plane.

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