Problem 54 - College Algebra

Question

Graph each equation in a rectangular coordinate system. \(x-0\)

Step-by-Step Solution

Answer

The graph of the equation \(x = 0\) is a vertical line that coincides with the y-axis.

Step 1: Understand the equation

The equation given is \(x = 0\). This equation is in the form of \(x = k\) where \(k\) is a constant. The graph of any equation in this form is a vertical line.

Step 2: Find the interception points with axes

For the equation \(x = 0\), the line intersects the x-axis at 0 and all other y-axis values. Thus, this line coincides with the y-axis.

Step 3: Graph the line

Plot the line \(x = 0\) on the rectangular coordinate system. The graph is a vertical line that coincides with the y-axis.

Key Concepts

Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian coordinate system, is a two-dimensional plane that allows us to graph equations using coordinates. This system consists of two axes:

• The horizontal axis, known as the x-axis.
• The vertical axis, known as the y-axis.

These two axes intersect at the origin, which is the point (0, 0). Each point on the plane is represented by a pair of numbers \(x, y\), where 'x' indicates the horizontal position and 'y' indicates the vertical position.

This system is essential for graphing linear equations, as it gives us a visual way to see the relationships between different variables.

Vertical Line

A vertical line in the rectangular coordinate system is a line that goes straight up and down. It does not tilt or lean. This kind of line has a special characteristic: all the points on the line have the same x-coordinate.

For instance, the equation \(x = 0\) describes a vertical line. This means no matter what y-value you choose, the x-value will always be 0. Consequently, for every point on this line, x remains constant while y can be any real number.

Vertical lines are unique because they cannot be described by the usual y = mx + b form, as their slope is undefined.

Interception Points

Interception points are where a line crosses the axes on the coordinate plane. In the equation \(x = 0\), the line intersects the x-axis at the origin (0,0) because that is the point where x is zero.

Since this is a vertical line, it does not cross the y-axis at any single point but rather runs directly along the y-axis itself. The points of intersection on the coordinate axes help in visualizing and graphing the line more accurately.

• On the x-axis: Intersection at (0,0).
• On the y-axis: Line coincides along the y-axis.

Equation of the Form x = k

Equations of the form \(x = k\) describe vertical lines where 'k' is a constant. Here, x has a fixed value, meaning for any point on this line, x always equals k, regardless of y.

For example, when given the equation \(x = 0\), this tells us that the line is vertical and passes through the x-coordinate of 0.

This form simplifies graphing since you only need to know the constant value of x to draw the line. It's an easy way to plot a straight vertical line on the coordinate plane.