91Ó°ÊÓ

The slope of a line is a key concept when graphing linear equations. It measures the steepness or incline of a line, and it is represented by the letter \( m \). The slope is calculated as the "rise" over the "run", which means how much the line moves up or down for a unit movement to the right.
The formula to calculate the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]In the original exercise, the slope is given as \( -\frac{1}{3} \). This indicates that for every 3 units you move right along the x-axis, you move 1 unit down on the y-axis. Here are some key points to understand about slope:

• A positive slope means the line rises as it moves from left to right.
• A negative slope, like \(-\frac{1}{3}\), means the line falls as it moves from left to right.
• If the slope is zero, the line is horizontal, indicating constant y-values.
• A undefined slope indicates a vertical line.

Understanding slope helps you determine the direction and angle of a line in the coordinate plane.

The coordinate plane is a two-dimensional surface where we can graph points, lines, and curves. It is defined by two axes: the x-axis (horizontal) and the y-axis (vertical). Where these axes intersect is known as the origin, denoted as \((0, 0)\).
On the coordinate plane, every point is represented by a pair of numbers, called coordinates, written as \((x, y)\). Here's a brief breakdown of the plane's features:

• The x-axis is the horizontal number line. Positive values extend to the right, while negative values extend to the left.
• The y-axis is the vertical number line. Positive values rise upward, and negative values fall downward.
• Quadrants: The plane is divided into four quadrants:
  • Quadrant I: both x and y are positive.
  • Quadrant II: x is negative, y is positive.
  • Quadrant III: both x and y are negative.
  • Quadrant IV: x is positive, y is negative.

The coordinate plane is essential for visualizing the graphs of equations. By plotting points and understanding their placement, we can draw lines representing linear equations.

Points on a line are crucial to defining the line’s path. To graph a linear equation, you need at least two points that lie on the line. Once you have these points, you can draw a straight line through them.
In our example, we started with the point \((3, -4)\) as given, and used the slope \(-\frac{1}{3}\) to find a second point, \((6, -5)\). Here's how you plot points:

• Identify the coordinates of the point, written as \((x, y)\).
• On the coordinate plane, start at the origin \((0, 0)\), move along the x-axis to the x-value.
• From this x-location, move vertically to reach the y-value.
• Mark the position with a dot.

Plotting multiple points helps you visualize the complete line that the equation represents. Once plotted, use a ruler to ensure the line is straight, extending it across the plane, ensuring that it reflects the line's equation.