Problem 50 - College Algebra

Question

Graph each equation in the rectangular coordinate system. $$x=5$$

Step-by-Step Solution

Answer

A line has been drawn in the rectangle coordinate system, where x is always 5.

Step 1: Analysing the equation

In the equation \(x=5\), observe that no y-values are mentioned, meaning for this line, x will always equal 5, regardless of what y is.

Step 2: Plotting a point

Start by plotting a point where \(x = 5\) on the x-axis. This point is (5,0).

Step 3: Drawing the line

Now, remember that it doesn't matter what y is for this line. So, this point at x=5 stretches both up and down the entire y-axis. Draw a vertical line extending both up and down from your point. Make sure the line passes through the correct point on the x-axis.

Key Concepts

Rectangular Coordinate System

Understanding the rectangular coordinate system is essential for graphing equations. It's a two-dimensional plane consisting of two perpendicular lines, known as axes. The horizontal axis is typically referred to as the x-axis, while the vertical axis is called the y-axis. These axes intersect at a point known as the origin, labeled as (0,0).
When you place a point on this system, it's given coordinates based on how far along each axis the point is located. For instance, the point (5, 0) lies 5 units to the right of the origin along the x-axis. In other words, the coordinates tell us the point's position relative to the origin. By using this system, we can plot points, lines, and curves to represent various mathematical relationships clearly.

Plotting Points

Plotting points is a basic skill that involves placing dots at specific coordinates on a rectangular coordinate system. To plot a point, you look at the coordinate pair: the first number tells you how far to move along the x-axis, and the second number tells you how far to move along the y-axis. The process is straightforward:

1. Begin at the origin (0,0) on the graph.
2. Move horizontally to the position indicated by the x-coordinate.
3. Then move vertically to the position indicated by the y-coordinate.
4. Mark the spot where these two movements meet - this is where you plot the point.

For example, to plot the point (5,0), start at the origin, move 5 units to the right along the x-axis since our x-coordinate is 5, and then don't move up or down because our y-coordinate is 0. Place a dot at this location to plot the point.

Vertical Lines

In coordinate geometry, vertical lines have a special equation format, typically written as x equals a constant value, like the equation in our example, which is x=5. These vertical lines are parallel to the y-axis and cross the x-axis at the point specified by the constant.
To graph a vertical line, you:

1. Plot a point where the line crosses the x-axis.
2. Then draw a straight line up and down from that point, extending it infinitely in both directions or as far as the graph requires.

The fact that such a line does not have a defined slope or y-coordinate might seem peculiar at first, but it's because the y-value is not restricted—the line extends throughout the entire y-axis. No matter what y-value you choose, as long as x is equal to 5, the point will lie on this vertical line.