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Chapter 8: Problem 21

Graph each equation. $$ x=6 $$

Short Answer

Expert verified
The graph is a vertical line at \(x=6\).

Step by step solution

01

Understand the Equation

The equation given is \(x = 6\). This tells us that for any point on the graph, the x-coordinate must be 6. This is a vertical line at \(x=6\) because the x-value doesn't change regardless of the y-value.
02

Plot Key Points

Since it's a vertical line, select points along this line. A few examples can be (6, 0), (6, 1), and (6, -1). These points have the same x-coordinate, 6, but different y-coordinates.
03

Draw the Line

On a graph, mark the points (6, 0), (6, 1), and (6, -1). Connect these points with a straight line. This line will be parallel to the y-axis and pass through all points where x is 6.
04

Extend and Verify

Extend the line upwards and downwards across the graph to ensure it's a straight vertical line. Verify that it doesn't curve and stays parallel to the y-axis.

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Key Concepts

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

Vertical Lines
Vertical lines on a graph are an important concept in linear equations.
These lines stand up straight and parallel to the y-axis. They occur when all points on the line share the same x-coordinate, but can have different y-coordinates.
For instance, the equation \( x = 6 \), describes a vertical line because every point across this line shares the x-value of 6.
Unlike horizontal or slanted lines, vertical lines demonstrate that the dependent variable (y) can be anything, while the independent variable (x) remains constant. This constancy of x makes vertical lines simple but powerful tools in various mathematical and real-world applications.
Coordinate Plane
A coordinate plane is like a two-dimensional map with horizontal and vertical axes.
The horizontal line is known as the x-axis, while the vertical line is called the y-axis. This plane is where every point is represented by a pair of numbers known as coordinates.
  • The first number in the pair is the x-coordinate, which tells you how far left or right the point is from the origin.
  • The second number is the y-coordinate, indicating how far up or down the point is from the origin.
The coordinate plane is crucial for graphing linear equations, such as \( x = 6 \), which describe relationships between x and y, and help show patterns visually on the graph.
Plotting Points
Plotting points is a foundational skill needed to graph lines accurately.
A point is defined by an ordered pair (x, y), representing its specific location on the coordinate plane. To plot a point:
  • Start at the origin (0,0).
  • Move horizontally to the x-value of the point.
  • Then move vertically to the y-value.
For the equation \( x = 6 \), you can select any y-value, such as 0, 1, or -1, and plot points like (6, 0), (6, 1), or (6, -1). These steps ensure your plotted points align correctly and form a vertical line on the graph.
Graph Interpretation
Interpreting a graph involves understanding what the visual representation states about the mathematical equation.
For example, a vertical line such as \( x = 6 \), tells us that no matter which point you choose along this line, the x-coordinate remains unchanged at 6.
This type of graph provides quick insight into how one variable affects another: the y-value may vary, yet x remains constant.In everyday contexts, graphs can show trends or relationships between two quantities, making it crucial to understand graph interpretations and their implications. Graphs turn complex data into a simpler, more comprehensive visual tool that can enhance decision-making.

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