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Chapter 7: Problem 104

Is the graph of the equation \(x=0\) the \(x\) -axis or the \(y\) -axis?

Short Answer

Expert verified
The graph of the equation \(x=0\) is the \(y\)-axis.

Step by step solution

01

Understand the equation

The equation given is \(x=0\). In this context, it means that for every point on the graph, the x-coordinate must be 0. This implies all points form a vertical line where x is constantly 0.
02

Identify graph behavior

Since the equation results in a vertical line, we need to determine which axis this line overlaps. The vertical line where all x-values are 0 does not cross any other points except those directly on the \( y \)-axis.
03

Determine the corresponding axis

A vertical line that runs where all x-values are 0 is the same as the \( y \)-axis. Therefore, \( x = 0 \) is the equation of the \( y \)-axis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Equations
Equations are mathematical statements that express the equality between two expressions. They consist of two sides, with an equal sign "=" indicating that both sides are equivalent.
In coordinate geometry, equations can represent lines on a graph.
One important type of equation you'll encounter is the linear equation, often taking the form \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept.With linear equations:
  • Changing the value of \( m \) alters the line's slope.
  • The y-intercept \( c \) determines where the line crosses the y-axis.
For our specific equation \( x = 0 \), the presence of only an \( x \) variable signifies that the y-coordinates can be any value, but the x-coordinate remains constant, producing distinct behavior on the graph.
Graphing Lines in the Coordinate System
When graphing lines in the coordinate plane, we use a set of \( (x, y) \) coordinates.
Each pair of \( x \) and \( y \) represents a point on the plane.To graph a line, knowing:
  • The equation provides a blueprint for plotting.
  • Linear equations like \( y = mx + c \) offer a direct way to plot by using the slope \( m \) and y-intercept \( c \).
For our equation \( x = 0 \), things are a bit different. This equation signifies a vertical line where the x-coordinate is always 0 regardless of y. To graph this:
  • Find the x-intercept (which is constantly 0).
  • Plot several points where \( x = 0 \) and different values of \( y \).
  • Connect these points with a straight line extending vertically.
This distinct equation skips the typical slope-intercept form \( y = mx + c \), making it a great example of a unique graphing scenario.
Exploring Vertical Lines
Vertical lines in coordinate geometry represent a unique class where only one variable is involved.
The equation of a vertical line takes the form \( x = a \), where \( a \) is a fixed value for \( x \).
Some key characteristics:
  • Vertical lines have an undefined slope.
  • They always run parallel to the y-axis.
  • No matter the value of y, the x-value remains constant.
For example, the line represented by \( x = 0 \) is special because it coincides directly with the y-axis.
In essence, this equation doesn't imply a typical slope as seen in other forms like \( y = mx + c \). Instead, it simply reflects a fixed position along the x-coordinate.
This characteristic makes vertical lines easy to identify yet crucial in understanding the intricacies of graph plotting and interpreting the behaviors of the coordinate plane.

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