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Chapter 1: Problem 43

Graph each ordered pair on a coordinate system. $$\gamma\left(2 \frac{3}{4}, 0\right)$$

Short Answer

Expert verified
Plot point \(\left(2 \frac{3}{4}, 0\right)\) on the x-axis between 2 and 3.

Step by step solution

01

Understand the Ordered Pair

An ordered pair is a pair of numbers used to locate a point on a coordinate system. It is written as \((x, y)\), where \(x\) is the horizontal position (x-coordinate) and \(y\) is the vertical position (y-coordinate). In this case, the ordered pair is \(\left(2 \frac{3}{4}, 0\right)\).
02

Convert Mixed Number to Improper Fraction (Optional)

If necessary, convert the mixed number to an improper fraction to make calculations easier. For the x-coordinate \(2 \frac{3}{4}\), it can be expressed as the improper fraction \(\frac{11}{4}\). However, this step is not strictly necessary for graphing.
03

Plot the X-Coordinate

On a coordinate system, locate the point on the x-axis by moving horizontally to \(2 \frac{3}{4}\). This is slightly less than 3. Imagine dividing the space between 2 and 3 into four equal parts and move three parts over.
04

Plot the Y-Coordinate

Since the y-coordinate is 0, the point is on the x-axis. This means you don't move vertically; your point remains on the horizontal line.
05

Mark the Point on the Coordinate System

On the graph, mark the point where the x value \(2 \frac{3}{4}\) meets the y value 0. This point is \(\left(2 \frac{3}{4}, 0\right)\) and should be plotted on the x-axis slightly to the left of 3.

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Key Concepts

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

Understanding the Coordinate System
A coordinate system is like a map that helps us find and describe locations. It consists of two perpendicular lines known as axes. The horizontal line is called the x-axis, and the vertical line is the y-axis. These axes intersect at a point called the origin, which is designated as (0, 0).
  • The coordinate system is crucial in mathematics for locating points in a plane.
  • It provides a way to describe exactly where a point is situated by using two numbers.
Each number in an ordered pair corresponds to one of these axes, and together, they identify a specific point on the plane.
The Role of the X-Coordinate
The x-coordinate is the first number in an ordered pair and determines the horizontal position of a point relative to the origin. In our example, the ordered pair is \((2 \frac{3}{4}, 0)\). The x-coordinate, \(2 \frac{3}{4}\), tells us how far to move along the x-axis from the origin.
  • Think of the x-coordinate as moving left or right from the origin.
  • If it's positive, you move to the right; if negative, you move to the left.
For the x-coordinate \(2\frac{3}{4}\), visualize dividing the space between the whole numbers 2 and 3 into four equal segments, then moving three segments over to position the point.
Understanding the Y-Coordinate
The y-coordinate is the second number in an ordered pair, which indicates the vertical position of a point relative to the origin. In our example, the y-coordinate is 0. This determines that no vertical movement is needed from the origin to find the point.
  • The y-coordinate tells you how far to move up or down from the origin.
  • A positive y-coordinate means moving up, while a negative one means moving down.
When the y-coordinate is 0, as in our example, it tells us that the point lies directly on the x-axis. This simplifies plotting, as we only need to consider the horizontal position.
Plotting Points on the Coordinate System
Plotting points involves two steps: considering the x-coordinate and the y-coordinate. For our ordered pair \((2\frac{3}{4}, 0)\):1. Start at the origin (0,0).2. Move horizontally to the point described by the x-coordinate \(2\frac{3}{4}\).3. Since the y-coordinate is 0, remain on the x-axis with no vertical movement.
  • Mark a small dot or cross at the position \((2\frac{3}{4}, 0)\).
  • The point should be located slightly to the left of 3 on the x-axis.
By following these steps, you can accurately locate any point given its coordinates, using the coordinate system.

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