Problem 65 - Introductory Algebra for College Students

Question

Graph each equation in a rectangular coordinate system. $$x=-2$$

Step-by-Step Solution

Answer

The graph of the equation \(x=-2\) is a vertical line passing through the point where x is -2 on the x-axis, and is parallel to the y-axis.

Step 1: Understanding the Equation

The equation is \(x=-2\). This is a vertical line equation. Vertical line equations have all x-coordinates as a constant value, which in this case is -2.

Step 2: Plotting the Line

Start by drawing a rectangular coordinate system. Since this is a vertical line, draw a straight line that passes through the point where x is -2 on the x-axis. The line will be parallel to the y-axis, meaning it extends indefinitely in both the positive and negative y-direction.

Key Concepts

Understanding the Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian coordinate system, is essential for graphing equations. It consists of two perpendicular lines known as axes. The horizontal axis is the x-axis, while the vertical axis is the y-axis. Together, they create a grid system that allows us to pinpoint any location on the plane using a pair of coordinates \(x, y\).

Here's how it works:

• Each point on the plane corresponds to an ordered pair \(x, y\).
• The origin, where the x-axis and y-axis intersect, is the point (0,0).
• x-coordinates tell you how far to move left or right from the origin, while y-coordinates show how far up or down to move.

Understanding this system is the foundation for plotting any graph, including vertical lines, within it.

Exploring Vertical Line Equations

A vertical line equation is one where x is constant, such as \(x = -2\). This means that no matter what the y-value is, x will always be -2. This is why vertical lines appear parallel to the y-axis.

Key characteristics of vertical lines:

• They have undefined slope since there's no horizontal change.
• The equation \(x = c\) represents a vertical line where x is always the value of c.
• They run perpendicular to horizontal lines, which have equations like \(y = c\).

By knowing this, you can easily determine that \(x = -2\) is a line traversing the y-axis at every y-coordinate while maintaining an x-coordinate of -2.

Plotting Vertical Lines

Plotting vertical lines, such as \(x = -2\), is straightforward once you understand the equation and coordinate system. Here's a simple guideline to do this:

• Locate the x-coordinate, in this case, -2, on the x-axis.
• Draw a straight line passing through the entire length of the grid parallel to the y-axis.
• This line will not cross the x-axis anywhere else except at \(x = -2\).

When plotting:

• Make sure the line extends in both directions—up and down—indicating it continues indefinitely.
• Use consistent scales on both axes for accuracy.
• Clearly label the line as \(x=-2\) to avoid any confusion.

By following these steps, you can accurately represent and understand vertical lines such as \(x = -2\) on the rectangular coordinate system.