Problem 60 - Intermediate Algebra

Question

Use the given conditions to determine in which quadrant of a rectangular coordinate system each point \((x, y)\) is located. \(x>0\) and \(y>0\)

Step-by-Step Solution

Answer

The point (x, y) with x>0 and y>0 is located in Quadrant I.

Step 1: Understanding the Coordinate System

In a rectangular coordinate system, also known as the Cartesian coordinate plane, there are four quadrants that are formed by two perpendicular lines called axes: the x-axis (horizontal) and the y-axis (vertical). The quadrants are numbered counterclockwise starting from the upper right quadrant.

Step 2: Identifying the Quadrants

The quadrants of the coordinate plane are designated as follows: Quadrant I is where both x and y are positive ( x>0, y>0 ), Quadrant II is where x is negative and y is positive ( x<0, y>0 ), Quadrant III is where both x and y are negative ( x<0, y<0 ), and Quadrant IV is where x is positive and y is negative ( x>0, y<0 ).

Step 3: Applying the Given Condition

Using the given conditions, where x>0 and y>0 , we look at which quadrants x and y belong to based on the signs of their values. Both x being positive and y being positive occurs only in Quadrant I.

Key Concepts

Cartesian coordinate system

The Cartesian coordinate system is central to many areas of mathematics and science. It provides a grid-like framework, formed by two perpendicular axes. The horizontal axis is the x-axis, and the vertical axis is the y-axis. These axes intersect at a point known as the origin, denoted by (0, 0).

Each point in this system is determined by a pair of numerical values, or coordinates, written as (x, y). The first number represents the position relative to the x-axis, while the second denotes the position relative to the y-axis.

For example, the point (3, 4) means moving 3 units along the x-axis and 4 units up the y-axis. This system allows for precise location of points within a two-dimensional plane.

quadrant identification

Knowing how to identify quadrants in the Cartesian coordinate system is crucial for understanding how points are located on a plane. The system divides the plane into four regions, called quadrants. They go counterclockwise starting from the upper right part of the plane.

Here's a simple way to remember them:

• **Quadrant I**: Both x and y values are positive, (x > 0, y > 0).
• **Quadrant II**: x-values are negative, y-values are positive, (x < 0, y > 0).
• **Quadrant III**: Both x and y values are negative, (x < 0, y < 0).
• **Quadrant IV**: x-values are positive, y-values are negative, (x > 0, y < 0).

Identifying which quadrant a point belongs to helps in understanding its location and behavior in various mathematical problems.

positive and negative values

Positive and negative values in the Cartesian coordinate system are essential in determining the position of points. These values indicate direction as well as distance from the origin.

**Positive Values**:

• For x-values, if x is positive, the point is to the right of the y-axis.
• For y-values, if y is positive, the point is above the x-axis.

**Negative Values**:

• For x-values, if x is negative, the point is to the left of the y-axis.
• For y-values, if y is negative, the point is below the x-axis.

Understanding these values is critical. They help identify not just the position but also the specific quadrant, which is vital for solving mathematical and scientific problems effectively.