Problem 53 - Algebra 1

Question

Plot the ordered pairs in a coordinate plane. $$(2,3),(-2,-3),(4,-2)$$

Step-by-Step Solution

Answer

To plot the ordered pairs, mark the points (2,3) in the first quadrant, (-2,-3) in the third quadrant, and (4,-2) in the fourth quadrant on a Cartesian plane.

Step 1: Understand the Cartesian Coordinate System

In a Cartesian coordinate system, points are defined by two coordinates (x, y). The x-coordinate represents the horizontal distance from the origin while the y-coordinate represents the vertical distance. The origin, where both coordinates are zero, is at the center of the grid. The coordinate plane is divided into four quadrants, labeled counterclockwise from top right: the first quadrant (x>0, y>0), second quadrant (x<0, y>0), third quadrant (x<0, y<0), and the fourth quadrant (x>0, y<0).

Step 2: Identify the Coordinates of the Points

The problem provides three points which will be plotted. They are (2,3), (-2,-3), and (4,-2). For each of these points, the first number is the x-coordinate and the second number is the y-coordinate.

Step 3: Plot the Points

To plot point (2, 3), start at the origin, move 2 units to the right along the x-axis (since x is positive) and 3 units up along the y-axis (since y is positive). So, the point (2, 3) is in the first quadrant. For (-2,-3), move 2 units to the left along the x-axis (since x is negative) and 3 units down along the y-axis (since y is negative), placing this point in the third quadrant. For (4,-2), move 4 units to the right along the x-axis (since x is positive) and 2 units down the y-axis (since y is negative), so this point lies in the fourth quadrant.

Key Concepts

Cartesian Coordinate System

The Cartesian Coordinate System is a mathematical grid used to locate points in a plane. This system consists of two perpendicular lines called axes. The horizontal line is the x-axis, and the vertical line is the y-axis. Where these two axes intersect is known as the origin, with coordinates (0, 0).

The system is named after the French mathematician René Descartes. It revolutionized mathematics by allowing algebra to be applied to geometry. In this grid, any point can be defined by an address or a set of coordinates that describe its position in space. These coordinates are written in ordered pairs like \(x, y\).

The Cartesian Coordinate System makes it easy to visualize and solve problems. For example, it allows us to plot shapes, find distances between points, and identify patterns or trends.

Ordered Pairs

Ordered pairs are crucial in describing the exact location of a point on the Cartesian plane. An ordered pair consists of two components: the x-coordinate and the y-coordinate. It is always written in the form \(x, y\).

• \(x\) - This represents the horizontal position on the x-axis.
• \(y\) - This represents the vertical position on the y-axis.

As the name "ordered pairs" suggest, the sequence matters. The first number always refers to the x-axis and the second to the y-axis. Mixing them up will lead to plotting the points wrongly. For example, the ordered pair (2, 3) means you travel 2 units along the x-axis (to the right) and then 3 units along the y-axis (upward).

This concept simplifies the process of locating points and helps in identifying spatial relationships in a statistical or graphical representation.

Quadrants

In a Cartesian Coordinate System, the plane is divided into four sections, known as quadrants. These quadrants help us in identifying the general position of points using the ordered pairs. They are numbered counterclockwise starting from the upper right quadrant.

• First Quadrant: Both x and y coordinates are positive (e.g., (2, 3)).
• Second Quadrant: x-coordinate is negative, and y-coordinate is positive (e.g., (-3, 4)).
• Third Quadrant: Both x and y coordinates are negative (e.g., (-2, -3)).
• Fourth Quadrant: x-coordinate is positive, and y-coordinate is negative (e.g., (4, -2)).

Each quadrant gives us quick knowledge about the sign of the coordinates of points located within it. It is an efficient way to understand and predict patterns or results in math problems and relate them to real-world contexts.