Problem 52 - Precalculus Essentials

Question

Graph equation in a rectangular coordinate system. $$x=5$$

Step-by-Step Solution

Answer

The graph of the equation \(x=5\) is a vertical line passing through points where the x-coordinate is 5.

Step 1: Identify the type of line

Since the equation is \(x=5\) where there is no \(y\) variable present, this is a vertical line.

Step 2: Plotting the line

In a rectangular coordinate system, plot a vertical line at x=5. This means the line passes through values where x is 5 on the x-axis. It will extend infinitely in both positive and negative y directions.

Step 3: Mark the points

You can mark some points on the line to make it clear that all the points on this line have x-coordinate as 5. For instance, points could be (5,0), (5, 2), (5,-2), (5,4) etc. All these points will lie on the line \(x=5\).

Key Concepts

Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian coordinate system, is a foundation for graphing. This system uses two perpendicular lines, usually denoted as the x-axis (horizontal) and the y-axis (vertical), to divide the plane into four regions called quadrants. Each point in this system can be described by an ordered pair, \((x, y)\). Here,

• x represents the horizontal position.
• y represents the vertical position.

When graphing, the intersection of these axes is called the origin and is designated by \((0, 0)\). The x-axis is aligned to increase from left to right, while the y-axis extends upwards and downwards. Being familiar with this setup is crucial for effectively locating and plotting points in graphing activities.

Vertical Line

A vertical line in a graph's rectangular coordinate system represents a constant x-value for all points on this line. In the equation \(x = 5\), the line is vertical because the value of x does not change regardless of the y-value. This contrasts with horizontal lines, which keep y constant while x can vary.Vertical lines have some unique characteristics:

• They are parallel to the y-axis.
• They do not have a slope, as their slope is considered undefined.
• All points have the same x-coordinate.

Such lines extend infinitely upwards and downwards, going through points like \((5, 1)\), \((5, -2)\), and any other pair where the x-coordinate is 5. Understanding how to graph vertical lines helps interpret equations without a y-term.

Plotting Points

Plotting points effectively on the rectangular coordinate system is crucial for drawing precise graphs. To plot a point, you need an ordered pair \((x, y)\). This pair tells you exactly where to place a point on the graph.Here's how you plot points:

• Start at the origin \((0, 0)\).
• Move x units along the x-axis. If x is positive, move right; if negative, move left.
• From that position, move y units parallel to the y-axis. Move up for positive y, down for negative y.

As an example, plotting the line \(x = 5\) involves just fixing your x-move to 5 and varying y to any value. Select some helpful points like \((5, 0)\), \((5, 3)\), and \((5, -4)\). All these points reinforce the concept of a vertical line at x = 5. Once plotted, these points illustrate the line visually, helping reinforce understanding the graphing process.