Problem 20 - Elementary Algebra

Question

For Problems 1-36, graph each linear equation. (Objective 2) $$ x=0 $$

Step-by-Step Solution

Answer

The graph of \(x = 0\) is a vertical line on the \(y\)-axis.

Step 1: Identify Equation Type

The given equation is \(x = 0\). This represents a vertical line, as all points on this line have the same \(x\)-coordinate, which is 0.

Step 2: Locate the Line on the Coordinate Plane

Since the equation is \(x = 0\), this line is located along the \(y\)-axis because this axis corresponds to where all \(x\) values are 0.

Step 3: Draw the Vertical Line

To graph the equation, draw a straight vertical line through the \(y\)-axis at \(x = 0\). This line extends infinitely in both the positive and negative directions along the \(y\)-axis.

Key Concepts

Coordinate Plane

The coordinate plane is a fundamental concept in graphing. It's a two-dimensional surface where you can plot points, lines, and curves.
These planes consist of two perpendicular number lines: the horizontal line called the x-axis and the vertical line called the y-axis.
The point where they intersect is known as the origin, with coordinates (0, 0).
Each point on the plane is defined by an ordered pair of numbers

• The first number is the x-coordinate, indicating how far to move left or right from the origin.
• The second number is the y-coordinate, indicating how far to move up or down.

The entire plane is divided into four sections called quadrants.
Starting from the top right and moving counter-clockwise, these are the first, second, third, and fourth quadrants.
Understanding this setup is crucial for graphing any equation, including linear ones.

Vertical Line

Vertical lines are unique features on a coordinate plane and are represented by equations of the form \(x = a\).
Here, "a" is a constant value, meaning it doesn't change, which defines the x-coordinate for every point on this line.
Let's examine why all points on a vertical line have the same x-coordinate:

• Consider the equation \(x = 0\). This indicates a vertical line where every point has an x-value of 0.
• Such a line runs parallel to the y-axis and essentially represents the y-axis itself when \(x = 0\).
• Unlike other lines, vertical lines don't have a well-defined slope, because their rise over run would result in division by zero.

Recognizing a vertical line helps in quickly sketching it on the coordinate plane. Simply identify the x-value, and draw a line parallel to the y-axis through where the x-coordinate remains constant.

Linear Equation Graphing Steps

Graphing a linear equation involves several straightforward steps. Let's break down these essential stages:

• **Identify the Equation Type**: Is it in the format \(y = mx + b\) or \(x = a\)? If it's \(x = a\), it indicates a vertical line.
• **Locate Key Points**: For a vertical line like \(x = 0\), the entire line runs along one x-coordinate. Here, it's the y-axis. Identifying key coordinates can help place the line accurately on the plane.
• **Graph the Line**: Draw the line through the necessary coordinates. For example, the equation \(x = 0\) is graphed by drawing a line parallel to the y-axis through the origin, extending both upward and downward.

This simple procedure helps in visualizing the solution quickly and aids in understanding how different types of lines appear on the coordinate plane.