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Chapter 7: Problem 61

What is the graph of \(x=0\) ? What is the graph of \(y=0\) ? Explain your answers.

Short Answer

Expert verified
The graph of \(x=0\) is a vertical line along the y-axis, and the graph of \(y=0\) is a horizontal line along the x-axis.

Step by step solution

01

Understanding the Equation

The equation given is \(x = 0\). This refers to a vertical line in the coordinate plane, where all points on this line have an x-coordinate of 0.
02

Graphing the Vertical Line

To graph \(x = 0\), draw a line through the y-axis where x=0. This will be a vertical line cutting through (0, y) for any y-value. The line will pass through points like (0,1), (0,2), (0,-1), etc.
03

Understanding the Equation

Now consider the equation \(y = 0\). This refers to a horizontal line in the coordinate plane, where all points on this line have a y-coordinate of 0.
04

Graphing the Horizontal Line

To graph \(y = 0\), draw a line along the x-axis where y=0. This will be a horizontal line cutting through the x-axis, passing through points such as (1,0), (2,0), (3,0), etc.
05

Visualizing the Graphs Together

If both equations are on the same graph, \(x = 0\) will be a vertical line intersecting the x-axis at the origin, while \(y = 0\) will be a horizontal line intersecting the y-axis at the origin. The intersection point of these lines is the origin (0, 0).

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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 coordinate plane are special kinds of lines that run straight up and down. They are defined by equations of the form \(x = a\), where \(a\) is a constant. This constant \(a\) tells us where the line crosses the x-axis. For example, the graph of \(x = 0\) is a vertical line that runs through all points where the x-value is 0, such as (0,1), (0,-2), and (0,5).

  • Every point on a vertical line has the same x-coordinate.
  • Vertical lines have undefined slope because they do not run horizontally.
  • They do not intercept the y-axis at any point other than where x=0, but extend infinitely vertically.
Understanding where these lines fall on the coordinate plane helps solve problems involving intersections and distance.
Horizontal lines
Horizontal lines stretch from left to right across the coordinate plane. They are defined by equations like \(y = b\), where \(b\) is a constant representing the y-coordinate for every point on the line. The line \(y = 0\) is a good example, where all points have a zero y-value, running through (1,0), (-5,0), and (100,0) along the x-axis.

  • Each point on a horizontal line has the same y-coordinate.
  • The slope of a horizontal line is zero, meaning it is completely flat.
  • This type of line intersects with the y-axis at the given y-coordinate \(y = b\), while going infinitely left and right.
Horizontal lines are essential for graphing systems of equations and understanding relationships within the plane.
Coordinate plane
The coordinate plane is a two-dimensional surface where we graph points, lines, and other geometric figures. It is divided into four parts by two axes: the x-axis (horizontal) and the y-axis (vertical).

  • The x-axis runs horizontally, adding positive values to the right of the origin and negative to the left.
  • The y-axis runs vertically, with positive values above and negative values below the origin.
  • The origin is the center point of the plane, located at (0,0), where the x-axis and y-axis intersect.
Using the coordinate plane allows us to visualize mathematical concepts such as functions, lines, and intersections. It helps in analyzing the relationship between different algebraic expressions and equations.
Intersection of lines
An intersection is where two or more lines meet on the coordinate plane. For example, the intersection of the vertical line \(x=0\) and the horizontal line \(y=0\) is the origin (0,0). This point is where both lines cross each other.

  • Intersections signify points that satisfy the equations of both lines.
  • If two lines intersect at a single point, that point's coordinates can simultaneously satisfy both lines' equations.
  • The concept of intersection is fundamental in solving systems of linear equations, as it can indicate unique solutions.
Understanding intersections provides insight into how equations relate graphically and how their solutions manifest on the plane.

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