91Ó°ÊÓ

The slope of a line is a measure of its steepness and direction.
It's represented by the letter 'm'. A slope can be positive, negative, zero, or undefined.
Here, we have a slope of \(-\frac{1}{3}\) which tells us a few key things:

• The negative sign indicates the line slopes downward as it moves from left to right.
  
• The fraction \(-\frac{1}{3}\) means the line falls 1 unit vertically for every 3 units it moves horizontally.
  

The formula for calculating slope between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) on a line is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \] This property helps us sketch the line easily once we have a point and the slope.

The Cartesian plane, also known as the coordinate plane, consists of two perpendicular lines called the x-axis (horizontal) and the y-axis (vertical).
These axes divide the plane into four quadrants. Each point on the plane is represented by an ordered pair (x, y).
Here's a quick overview:

• The point where the axes intersect is called the origin, denoted as (0, 0).
  
• Points to the right of the origin have positive x-values and points to the left have negative x-values.
  
• Points above the origin have positive y-values and points below have negative y-values.
  

When plotting points or drawing lines, it's essential to understand this grid system to locate and connect points accurately.

Plotting points is a fundamental skill in graphing linear equations.
It involves marking a point on the Cartesian plane based on its coordinates (x, y).
Let's go through the steps for plotting a point:

• Identify the x-coordinate (horizontal position) and move that many units left or right from the origin.
  
• Next, identify the y-coordinate (vertical position) and move that many units up or down from your x-position.
  
• Mark the point where these coordinates meet.
  

In our exercise, we first plotted the point (1, -1). Moving right 1 unit and down 1 unit from the origin helps us locate this point.
Then, using the slope \(-\frac{1}{3}\), we moved 3 units right and 1 unit down to plot the second point (4, -2).

A linear equation represents a straight line when graphed on the Cartesian plane.
The general form of a linear equation is: \[ y = mx + b \]
Here, 'm' is the slope and 'b' is the y-intercept (the point where the line crosses the y-axis).
Given the slope and a point on the line, we can derive the equation by substituting the values into the formula.
For example, with slope \(-\frac{1}{3}\) and point (1, -1), the equation of the line becomes:
\[-1 = -\frac{1}{3}(1) + b\]
Solve for 'b' to find the y-intercept.
Thus, the complete equation of our line is:
\[ y = -\frac{1}{3}x - \frac{2}{3} \]
Understanding linear equations allows us to graph lines accurately and analyze their relationships.