Problem 27 - Understanding Elementary Algebra with Geometry

Question

In Exercises \(21-32,\) indicate which quadrant contains the given point. If a point lies on one of the coordinate axes, indicate which one. $$(-2.6,4.9)$$

Step-by-Step Solution

Answer

Quadrant II

Step 1: Understand the Cartesian Coordinate System

In the Cartesian coordinate system, there are four quadrants. Quadrant I is where both values (x, y) are positive. Quadrant II is where x is negative and y is positive. Quadrant III is where both values are negative. Quadrant IV is where x is positive and y is negative.

Step 2: Identify the Sign of the Coordinates

Given point is \((-2.6, 4.9)\). The x-coordinate is \(-2.6\), which is negative. The y-coordinate is \(\text{4.9}\), which is positive.

Step 3: Determine the Quadrant

Since the x-coordinate is negative and the y-coordinate is positive, the point \((-2.6, 4.9)\) lies in Quadrant II.

Key Concepts

Understanding Quadrants in the Coordinate System

The Cartesian coordinate system divides the plane into four quadrants using two perpendicular lines called axes. The horizontal axis is the x-axis, and the vertical axis is the y-axis. These axes intersect at the origin, which is \((0,0)\). Let me explain the four quadrants:

• Quadrant I: Both x and y coordinates are positive (\((+, +)\)).
• Quadrant II: x is negative and y is positive (\((-,-)\)).
• Quadrant III: Both x and y coordinates are negative (\((-,-)\)).
• Quadrant IV: x is positive and y is negative (\((+, -)\)).

These quadrants help us locate any point in the plane by identifying the signs of its coordinates.

Identifying Coordinates in the Cartesian Plane

In the Cartesian coordinate system, every point is identified by a pair of numbers known as coordinates. Coordinates are written in the form \((x, y)\). The x-coordinate tells us the point's horizontal position, while the y-coordinate tells us the point's vertical position. For instance, in the point \((-2.6, 4.9)\), -2.6 is the x-coordinate and 4.9 is the y-coordinate.

When identifying coordinates, it is crucial to note:

• If x is positive, it is to the right of the y-axis.
• If x is negative, it is to the left of the y-axis.
• If y is positive, it is above the x-axis.
• If y is negative, it is below the x-axis.

Determining Quadrants Based on Coordinates

To determine which quadrant a point lies in, we need to look at the signs of its coordinates. Let's use the given point \((-2.6, 4.9)\) as an example.

• The x-coordinate is -2.6, which is negative.
• The y-coordinate is 4.9, which is positive.

Since the x-coordinate is negative and the y-coordinate is positive, the point lies in Quadrant II. In this quadrant, coordinates always have a negative x value and a positive y value.

Remember this easy rule:

• If both x and y are positive, the point is in Quadrant I.
• Negative x and positive y points to Quadrant II.
• If both coordinates are negative, the point is in Quadrant III.
• Positive x and negative y coordinates indicate Quadrant IV.