Problem 51 - College Algebra Essentials

Question

Graph each equation in a rectangular coordinate system. $$x=-3$$

Step-by-Step Solution

Answer

The graph of the equation \(x = -3\) is a vertical line at \(x = -3\)

Step 1: Understand the equation

The equation given is \(x = -3\). This equation represents a vertical line on the graph as \(x\) is constant for all values of \(y\).

Step 2: Prepare to Plot

Setup a rectangular coordinate system. Since the equation does not stipulate any constraints on the value of \(y\), this line will extend indefinitely in both the positive and negative \(y\) directions.

Step 3: Plot the Line

Draw a vertical line at \(x = -3\) on the plane. This line represents all the solutions to the given equation. Every point on this line satisfies the equation.

Key Concepts

Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian coordinate system, is a two-dimensional plane defined by a pair of perpendicular axes: the horizontal axis (usually labeled as the x-axis) and the vertical axis (usually labeled as the y-axis). This system is used to graph equations, plot points, and represent relationships between two variables.

Each point on the plane is defined by an ordered pair \(x, y\), where \x\ represents the horizontal position and \y\ represents the vertical position. Together, the intersection of these axes forms the origin, which is denoted by the point \(0, 0\).

This coordinate system allows for various mathematical operations, such as measuring distances between points, determining slopes of lines, and solving geometric problems. It is crucial for visualizing mathematical equations and understanding their graphical representations.

Vertical Lines

A vertical line on a graph is a line that runs straight up and down across the Cartesian plane, maintaining a constant x-coordinate. In the context of graphing equations, a common equation form of a vertical line is \(x = c\), where \c\ is a constant.

• This means for any point on that line, the x-value remains unchanged, while the y-value can vary.
• For instance, the equation \(x = -3\) represents a vertical line where every point on this line has an x-coordinate of -3, regardless of the y-coordinate.

Vertical lines have interesting properties:

- **Undefined Slope**: The slope of a vertical line is undefined because division by zero occurs when trying to compute slope using the formula \(m = \frac{\Delta y}{\Delta x}\) - the change in y over the change in x.

- **Parallel to the y-axis**: All vertical lines are parallel to the y-axis since they never touch or cross each other.

Understanding vertical lines is important for grasping how specific equations can be visualized on the coordinate plane.

Plotting Graphs

Plotting graphs involves visually representing equations or data sets on a coordinate plane to analyze relationships and identify patterns. In mathematical graphs, points are plotted based on their coordinates, and lines or curves are drawn to represent equations.

To plot an equation like \(x = -3\):

• Begin by setting up a Cartesian plane, ensuring that both axes are clearly labeled.
• Locate the x-coordinate of -3 on the horizontal axis.
• Draw a straight vertical line passing through \(x = -3\), extending infinitely in both directions of the y-axis.

This visual process allows one to see the entirety of solutions for the given equation, as every point along the line \(x = -3\) is a solution. Graphing equations helps in understanding different scenarios, such as intercepts, slopes, and intersections. By plotting graphs, one can solve complex equations, identify trends, and make predictions based on graphical data.