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Chapter 1: Problem 11

Determine the quadrant(s) in which \((x, y)\) is Iocated so that the condition(s) is (are) satisfied. $$x=-4 \text { and } y>0$$

Short Answer

Expert verified
The coordinates are located in Quadrant 2.

Step by step solution

01

Identify Position of X and Y

Given x=-4 which is less than 0 and y>0 which is greater than 0, thus y is positive and x is negative.
02

Compare Conditions with Quadrants

Inspecting the rules of Cartesian coordinates: Quadrant 2 is the only quadrant where x is negative and y is positive.
03

Conclusion

Therefore, the coordinates (x, y) with x=-4 and y>0 are located in Quadrant 2.

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Key Concepts

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

Quadrants of the Cartesian Plane
In a Cartesian coordinate plane, the space is divided into four regions, known as quadrants. These quadrants help in identifying the position and characteristics of points based on their coordinates.
  • Quadrant 1: Here both coordinates are positive, i.e., \( x > 0 \) and \( y > 0 \).
  • Quadrant 2: In this quadrant, the x-coordinate is negative and the y-coordinate is positive, i.e., \( x < 0 \) and \( y > 0 \).
  • Quadrant 3: This quadrant has both coordinates negative, i.e., \( x < 0 \) and \( y < 0 \).
  • Quadrant 4: In the fourth quadrant, the x-coordinate is positive and the y-coordinate is negative, i.e., \( x > 0 \) and \( y < 0 \).
When working with coordinates like \( x = -4 \) and \( y > 0 \), we look for the quadrant where the x value is negative and the y value is positive, which is Quadrant 2. Understanding these quadrants is essential for graphing and interpreting points on the Cartesian plane.
Coordinate System
A coordinate system allows us to pinpoint the location of points within different sections of space. The Cartesian coordinate system is one of the most famous,
providing a way to describe locations using pairs of numbers:
  • X-coordinate (abscissa): This number tells us how far left or right a point is from the vertical axis (y-axis).
  • Y-coordinate (ordinate): This number indicates how far up or down a point is from the horizontal axis (x-axis).
For example, in the coordinate pair \( (-4, y) \),
-4 is the X-coordinate that indicates the point is 4 units to the left of the y-axis. The value of the Y-coordinate (\( y > 0 \)) suggests the point is above the x-axis. By plotting these coordinates on a graph, we can determine their exact location including which quadrant they reside in.
Integer Coordinates
Integer coordinates are points on the Cartesian plane where both x and y values are whole numbers. Points like \( (-4, y) \) are part of this practical approach, which simplifies many mathematics tasks. Here's why integer coordinates are important:
  • Simplicity: Using whole numbers makes calculations easier and decreases chances of error.
  • Graphing: Integer points can be precisely plotted on a standard grid without fractions or decimals, simplifying graphing tasks.
    They provide a straightforward visualization of positions.
  • Clear Quadrant Identification: With integer coordinates, identifying the quadrants becomes easier. If an integer coordinate has a negative x and positive y, like \( (-4, y) \), it's immediately clear that the point is in Quadrant 2.
Overall, integer coordinates are a staple in mathematics for their practicality and ease of use in diverse applications.

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