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Chapter 3: Problem 84

Answer each exercise with true or false. Point \((3,0)\) lies on the \(y\) -axis.

Short Answer

Expert verified
False.

Step by step solution

01

Understand the Coordinate System

The coordinate system consists of two axes, the x-axis (horizontal) and the y-axis (vertical). To determine if a point lies on the y-axis, examine its x-coordinate.
02

Analyze the Given Point

The point given is (3, 0). This point has an x-coordinate of 3 and a y-coordinate of 0.
03

Determine Location Relative to the y-axis

A point lies on the y-axis if its x-coordinate is 0. Given this point (3, 0), the x-coordinate is not 0; it is 3.

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Key Concepts

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

Understanding the x-axis
In the coordinate system, the x-axis runs horizontally from left to right across the plane. It is essential in defining the position of points in a two-dimensional space.
Imagine drawing a number line horizontally. This is your x-axis. Points are positioned on the x-axis using their x-coordinates.
Here's how it works:
  • The x-coordinate tells you how far left or right a point is from the origin, which is the center of the coordinate plane typically labeled as (0,0).
  • On the x-axis, as you move to the right of the origin, the values of the x-coordinates increase positively.
  • As you move to the left, the x-coordinates decrease, making them negative.
This horizontal axis is critical for plotting and reading coordinates in mathematical graphs and helps in interpreting and solving problems involving geometric figures and algebraic equations.
Grasping the y-axis
The y-axis is the vertical number line in the coordinate system, cutting through the origins and extending upward and downward.
The y-axis serves to measure distances vertically in a plane.
Here's what you need to know:
  • The y-axis is used when interpreting the second part of a coordinate pair (x, y).
  • Points that lie directly on the y-axis will have an x-coordinate of 0 because they do not move left or right from the origin.
  • Move up from the origin, the values of y-coordinates increase positively; move down, and these values decrease into negatives.
When analyzing a point like (3,0), understanding the y-coordinate is crucial to assess its vertical positioning. In this point, 0 indicates it neither moves up nor down from the x-axis, remaining at the base of the y-axis.
Significance of the x-coordinate
The x-coordinate plays a vital role in determining a point's location concerning the x-axis and other points within the plane.
It is usually the first number in an ordered pair (x, y).
Let's delve into its importance:
  • It tells you how far and in which direction to move horizontally from the origin.
  • If a point has an x-coordinate of 0, it means the point is located on the y-axis, as it involves no lateral movement.
  • In the example point (3,0), the x-coordinate being 3 signifies the point's distance from the y-axis horizontally.
So, in our case, since the x-coordinate is not 0, the point (3,0) does not lie on the y-axis, validating that this statement is false. Understanding x-coordinates is crucial in various applications, from navigating maps to developing computer graphics.

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Most popular questions from this chapter

Write the equations of three lines parallel to \(10 x-5 y=-7\) (See the third Concept Check.)

Given the points \((2,3)\) and \((-5,1),\) can the slope of the line through these points be calculated by \(\frac{1-3}{2-(-5)} ?\) Why or why not? (See the second Concept Check.)

See Examples 1 through 7 . Find an equation of each line described. Write each equation in slope-intercept form (solved for \(y\) ), when possible. Slope \(\frac{3}{5},\) through \((4,4)\)

Determine whether each pair of lines is parallel, perpendicular, or neither. \(x-3 y=-6\) \(y=3 x-9\)

See Examples 1 through 7 . Find an equation of each line described. Write each equation in slope-intercept form (solved for \(y\) ), when possible. With slope 0, through \((6.7,12.1)\)
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