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Chapter 2: Problem 52

Graph each equation in a rectangular coordinate system. $$ x=5 $$

Short Answer

Expert verified
The graph of the equation \(x = 5\) is a vertical line passing through the point (5,0)

Step by step solution

01

Understand the equation

The given equation is \(x = 5\). This equation does not involve any \(y\) term, which means the line will not slope in the y-direction. Hence, it's a vertical line.
02

Plot the line in the coordinate system

Start by marking the point (5,0) because the line \(x = 5\) passes through this point. Now, draw a straight vertical line through this point. The line should continue forever in both the upward and downward direction, because there is no restriction on the value of \(y\).

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Key Concepts

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

Rectangular Coordinate System
The rectangular coordinate system, also known as the Cartesian plane, is a two-dimensional plane used for graphing equations. It consists of two perpendicular number lines that intersect at a point called the origin. These lines are referred to as:
  • The x-axis (horizontal axis)
  • The y-axis (vertical axis)
The origin is located at the coordinate (0,0). This system allows us to locate any point within the plane using a pair of numbers known as coordinates, which are written as (\(x, y\)). These coordinates tell us how far a point is from the origin along each axis. The rectangular coordinate system is fundamental to plotting equations and analyzing their behavior.
Vertical Line
A vertical line on a rectangular coordinate system is a straight line that runs up and down parallel to the y-axis. Its defining feature is that every point on the line has the same x-coordinate. For the equation \(x = 5\), all points have an x-coordinate of 5. This means:
  • The line is consistent and does not slant with respect to the x-axis.
  • There is no restriction on the y-coordinate values, implying that the line extends infinitely in both upward and downward directions.
Vertical lines are special because they are the only lines not represented by the function form \(y = mx + b\) since slope \(m\) in this case would be undefined.
Plotting Points
Plotting points involves placing points on the rectangular coordinate system at specific \((x, y)\) coordinates. To plot the point (5, 0) from the equation \(x = 5\):
  • Start at the origin \((0,0)\).
  • Move 5 units along the x-axis to the point where x = 5.
  • No movement is required along the y-axis because y = 0.
Once this point is marked, you can then draw the vertical line passing through all points where x equals 5. Plotting points is a crucial skill in coordinate geometry because it helps in drawing accurate graphs and understanding the positioning of lines.
Coordinate Geometry
Coordinate geometry, often called analytic geometry, is the study of geometry using a coordinate system. Using numbers, we can describe geometric figures and analyze their properties. In this context:
  • Equations represent geometric figures such as lines, circles, and parabolas.
  • The line \(x = 5\) is one such figure, representing a vertical line on the coordinate plane.
Coordinate geometry allows for the precise graphing and interpretation of these figures. It bridges algebra and geometry, enabling complex geometric problems to be solved using algebraic equations and methods. Understanding how to graph equations and interpret lines is foundational for further studies in mathematics.

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Most popular questions from this chapter

Does \((x-3)^{2}+(y-5)^{2}=0\) represent the equation of a circle? If not, describe the graph of this equation.

What is a circle? Without using variables, describe how the definition of a circle can be used to obtain a form of its equation.

Give the center and radius of the circle described by the equation and graph each equation. Use the graph to identify the relation鈥檚 domain and range. $$(x+4)^{2}+(y+5)^{2}=36$$

Complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. $$x^{2}+y^{2}+3 x-2 y-1=0$$

A department store has two locations in a city. From 2012 through \(2016,\) the profits for each of the store's two branches are modeled by the functions \(f(x)=-0.44 x+13.62\) and \(g(x)=0.51 x+11.14 .\) In each model, \(x\) represents the number of years after \(2012,\) and \(f\) and \(g\) represent the profit, in millions of dollars. a. What is the slope of \(f ?\) Describe what this means. b. What is the slope of \(g ?\) Describe what this means. c. Find \(f+g .\) What is the slope of this function? What does this mean?
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