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Chapter 3: Problem 73

Given that \(x\) is a positive number and that \(y\) is a positive number, determine the quadrant or axis in which each point lies. $$ (-x,-y) $$

Short Answer

Expert verified
The point (-x, -y) is in Quadrant III.

Step by step solution

01

Understand the Quadrants

The coordinate plane is divided into four quadrants. Quadrant I has both x and y positive. Quadrant II has x negative and y positive. Quadrant III has both x and y negative. Quadrant IV has x positive and y negative. The axes are not considered part of any quadrant.
02

Check Sign of -x and -y

Given the point (-x, -y) , where both x and y are positive, -x is negative and -y is also negative.
03

Determine Location Based on Negatives

Since both coordinates are negative, the point (-x, -y) is located in Quadrant III.

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Key Concepts

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

Quadrants
The coordinate plane is a two-dimensional surface where we plot points using pairs of numbers, specifically coordinates (\(x\) and \(y\)). To better organize this surface, imagine it being divided into four sections called quadrants. These quadrants help us quickly identify the sign of coordinates and where a point will lie at a glance.

  • Quadrant I: Both \(x\) and \(y\) are positive.
  • Quadrant II: \(x\) is negative, and \(y\) is positive.
  • Quadrant III: Both \(x\) and \(y\) are negative.
  • Quadrant IV: \(x\) is positive, and \(y\) is negative.
It's crucial to remember that the axes, the horizontal and vertical lines (\(x\)-axis and \(y\)-axis), are not considered parts of any quadrant. This division into quadrants is fundamental in coordinate geometry as it helps us categorize and understand the behavior of plotted points.
Positive and Negative Numbers
On the coordinate plane, numbers can be either positive or negative, and this determines their position relative to the origin (the center point, where the \(x\)-axis and \(y\)-axis intersect). Understanding the nature of these numbers is key to identifying which quadrant a point falls in.

Positive numbers move us away from the origin in different directions depending on the axis.
  • For the \(x\)-coordinate, positive values extend to the right of the origin, while negative values extend to the left.
  • For the \(y\)-coordinate, positive values rise above the origin, while negative values drop below.
To solve problems involving coordinates, determine the sign of each number. Points with both negative \(x\) and \(y\) coordinates, like (-x, -y), are found in Quadrant III since negative movements occur to the left and down from the origin.
Axes in Coordinate System
The axes in the coordinate system are pivotal reference lines that make it possible to discuss positions in terms of coordinates. They help us split the two-dimensional plane into distinct regions, evident in concepts such as quadrants.

  • The \(x\)-axis is the horizontal line that separates the plane into top and bottom halves.
  • The \(y\)-axis is the vertical line that splits the plane into left and right halves.
Both axes intersect at the origin, marked as point (0, 0), and neither is part of any quadrant. They serve as reference points, providing a baseline for both positive and negative movements along the plane. By familiarizing ourselves with the axes, we gain a better understanding of how to navigate each quadrant effectively, determining the location and nature of points based on their coordinates.

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