Problem 52 - Introductory Algebra for College Students

Question

Graph equation. \(x=4\)

Step-by-Step Solution

Answer

The graph of the equation \(x=4\) is a vertical line passing through the point (4,0) on the x-axis.

Step 1: Understand the Equation

The equation \(x=4\) means that for each point on the line, the x-coordinate is 4 and the y-coordinate can be any real number. This type of equation describes a vertical line in Cartesian coordinate system.

Step 2: Plot the Line

We need a Cartesian coordinate system to graph the line. Draw a vertical line that passes through the point (4,0) on the x-axis. This line represents all the points where the x-coordinate is 4, fitting the given equation \(x=4\).

Key Concepts

Cartesian Coordinate System

The Cartesian coordinate system is a fundamental framework for graphing equations that provides a visual representation of algebraic statements. It consists of two perpendicular number lines: the horizontal x-axis and the vertical y-axis. Each point in the coordinate system is defined by an ordered pair (x, y) where 'x' represents the position along the x-axis and 'y' represents the position along the y-axis.

To plot the line for the equation provided in our exercise, we utilize this system by identifying points that satisfy the equation and marking them on the graph. The intersection of these two axes is known as the origin, denoted as (0,0), which is the starting point for measuring distances along the axes.

Vertical Lines in Algebra

In algebra, vertical lines have a distinctive characteristic: all points on the line share the same x-coordinate. The general equation for a vertical line in algebra is of the form \(x=a\), where 'a' represents a constant, and it indicates that no matter what value 'y' takes, 'x' will always be 'a'. This is what makes it vertical – it is parallel to the y-axis.

For the specific equation \(x=4\), every single point has an x-coordinate of 4. This is independent of the y-value, which can vary from negative to positive infinity. Vertical lines go through all y-values but never traverse across different x-values. They are a perfect example of the function concept not applying, as each input x corresponds to many outputs y.

Plotting Points

When plotting points to graph an equation, one starts by identifying specific pairs (x, y) that satisfy the equation. For vertical lines, where x is a constant, we choose several values for y and maintain the same x for each. For instance, with the equation \(x=4\), we could select (4, -2), (4, 0), and (4, 3) as points meeting the criterion.

These points are plotted by starting at the origin, moving horizontally to the appropriate x-value, and then vertically to the corresponding y-value. By connecting these points with a straight edge, we form the graph of the equation – a vertical line intersecting the x-axis at x=4. The process of plotting points is critical as it lays the foundation for understanding more complex functions and interpreting graphs in algebra and other branches of mathematics.