Problem 51 - Introductory and Intermediate Algebra for College Students 4th

Question

Graph each equation. $$x=2$$

Step-by-Step Solution

Answer

The graph of the equation \(x = 2\) is a vertical line passing through the points with x-coordinate as 2.

Step 1: Understand the equation

The given equation is \(x = 2\). This is an equation of a vertical line that goes through the point (2,0). All points on this line have x-coordinate as '2'.

Step 2: Plotting the line

Start by drawing a set of axes. Mark the x-axis at 2. Now, as long as the x-coordinate is 2, y can be anything. This means our line will pass through (2,0), (2,1), (2,-1), etc. Because of that, we can draw a straight line that goes through all these points to create a vertical line at x=2 on the graph.

Step 3: Check the Graph

Once the graph is drawn, it is important to check our work. The line should be vertical and passing through the points where x-value is 2. This confirms the solution to the problem.

Key Concepts

Vertical Lines

Vertical lines are quite unique in how they are represented on a graph. Unlike other lines, which are often expressed in the form of \( y = mx + b \), vertical lines have equations that look like \( x = a \), where \( a \) is a constant. This means that every single point on a vertical line has the same x-coordinate, which in our exercise, is x=2. Thus, irrespective of the y-value, the x-value will remain consistent along the vertical line.
Vertical lines are particularly interesting because they defy the common slope-intercept form used for most straight lines. In fact, vertical lines are said to have undefined slope, as the rise over run (abla y / abla x) results in division by zero. This is integral to their nature, ensuring they run parallel to the y-axis and offer a clear visualization of constant x-values.

Coordinate System

The coordinate system is fundamental to graphing any linear equation, including vertical lines. It is usually made up of two perpendicular lines or axes called the x-axis and y-axis. These axes divide the plane into four quadrants. Points are plotted with coordinates, consisting of an x-value and a y-value, written as (x, y).

• The x-axis is horizontal, running left to right. Positive numbers are to the right of the origin (0,0), while negative numbers are to the left.
• The y-axis is vertical, running from top to bottom. Positive numbers are above the origin, and negative numbers are below.

Understanding the coordinate plane helps with correctly plotting vertical lines such as \( x = 2 \). The x-coordinate remains constant because the line runs up and down parallel to the y-axis, showing that all points share the same x-value.

Graphing Techniques

Effective graphing techniques simplify the process of plotting different types of lines, including vertical lines. For a line such as \( x = 2 \), start by carefully drawing a set of axes. Making sure they are labeled correctly is crucial for accuracy.
When graphing a vertical line, follow these steps:

• Locate the x-coordinate indicated by the equation, which is x=2 for our example. Mark this point unmistakably on the x-axis.
• Since any y-value is possible, draw a straight line through the point on the x-axis (2,0) extending vertically in both directions.
• Always check your graph by verifying that the line crosses all points where x is equal to 2, ensuring your graph is accurate and aligns with the given equation.

Such techniques aid in creating precise and reliable graphs, crucial for interpreting data and understanding linear relationships.