91Ó°ÊÓ

The Cartesian plane is a foundational concept in geometry and algebra that helps us locate points and visualize relationships between them. It consists of two perpendicular lines called axes: the horizontal axis, known as the x-axis, and the vertical axis, known as the y-axis. These axes intersect at a point called the origin, denoted as (0,0).

By plotting points on this grid, we can easily describe and analyze geometric figures and algebraic relationships. Each point on the Cartesian plane is defined by a pair of numbers, referred to as coordinates. These coordinates tell us how far along the x-axis and y-axis the point is located.

• The first number in the coordinate pair being the x-coordinate.
• The second number being the y-coordinate.

Coordinates are usually indicated as (x, y). Understanding this system allows us to effectively communicate exact positions on the plane.

Plotting points on the Cartesian plane is a straightforward process once you understand the coordinate system. Each ordered pair of coordinates corresponds to a unique point on the plane.

• To plot the point (6,0), start at the origin.
• Move 6 units right along the x-axis.
• The y-coordinate is 0, so you do not move vertically.

This places your point on the x-axis.

Next, for the point (0,4), begin again at the origin. This time, you move 0 units along the x-axis, but go 4 units up along the y-axis. Now you've plotted two points on the plane. By connecting these points with a straight line, you represent the set of all possible points that line up between these coordinates.

The concept of slope is key to understanding how steep a line is between two points. The slope is often represented by the letter m. The slope of a line gives insight into how much the line rises vertically compared to its horizontal run. It's calculated using the slope formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
This formula takes two points, (x_1, y_1) and (x_2, y_2), and finds the difference in their y-coordinates, divided by the difference in their x-coordinates.

In the example given:

• The first point is (0,4) and the second is (6,0).
• Apply the slope formula: \[ m = \frac{4 - 0}{0 - 6} \]
• This simplifies to \[ m = - \frac{2}{3} \].

The negative sign indicates a line that slopes downwards as it moves from left to right across the plane. The magnitude 2/3 tells us that for every 3 units traveled horizontally, the line falls by 2 units vertically.