91Ó°ÊÓ

The Cartesian plane is a two-dimensional coordinate system that helps us graph points, lines, and curves. It's defined by two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). The point where these axes intersect is called the origin and has coordinates (0, 0). Each point on the plane can be described with an ordered pair of numbers \(x, y\), where x indicates the position along the x-axis and y indicates the position along the y-axis.
When plotting points, always start from the origin. First, move horizontally to the x-coordinate, then vertically to the y-coordinate. For example, to plot the point \((-2, -4)\), move 2 units to the left and then 4 units down from the origin.
Understanding the Cartesian plane is fundamental for graphing linear equations, as it allows us to visualize and interpret solutions easily.

The slope of a line measures how steep the line is. It's a crucial concept in linear equations and can be found using the slope formula. The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[m = \frac{(y_2 - y_1)}{(x_2 - x_1)}\]
This formula calculates the 'rise' (the vertical change) over the 'run' (the horizontal change) between the two points. For instance, given points \((-2, -4)\) and \((-5, -2)\), substituting into the formula, we get:
\[m = \frac{(-2 - (-4))}{(-5 - (-2))} = \frac{2}{-3} = -\frac{2}{3}\]
A negative slope indicates the line descends from left to right, while a positive slope indicates it ascends.

Plotting points on a Cartesian plane is the foundation of graphing linear equations. To plot points, we need their coordinates \(x, y\). For our exercise, we are given \((-2, -4)\) and \((-5, -2)\).
Steps to plot points:

• Begin at the origin (0, 0).
• For \((-2, -4)\), move 2 units left along the x-axis and 4 units down along the y-axis. Mark this point.
• For \((-5, -2)\), move 5 units left along the x-axis and 2 units down along the y-axis. Mark this point.

Once plotted, you can draw a straight line through these points, representing all the solutions to the linear equation passing through them.

The phrase 'rise over run' is a simple yet effective way to remember how to calculate the slope of a line. 'Rise' refers to the change in y-coordinates (vertical movement), and 'run' refers to the change in x-coordinates (horizontal movement).
For the given points \((-2, -4)\) and \((-5, -2)\):

• Rise: Calculate the difference in y-values: \(-2 - (-4) = 2\)
• Run: Calculate the difference in x-values: \(-5 - (-2) = -3\)

Thus, the slope \((m)\) is \(\frac{rise}{run} = \frac{2}{-3} = -\frac{2}{3}\)
This negative slope tells us that for every 2 units the line rises, it moves 3 units to the left. Understanding 'rise over run' helps in graphing and interpreting lines.