An illustration shows a coordinate plane with the x-axis ranging from negative 9 to 9 in one unit increments, and the y-axis ranging from negative 11 to 11 in one unit increments. A solid line passes through two plotted points and extends beyond. A solid line with arrows at both ends passes through points with coordinates left parenthesis 0 comma 0 right parenthesis and left parenthesis 6 comma 6 right parenthesis.

Question

Explain why the slope is positive or negative.

Answer 1

To determine why the slope of the line passing through the points (0, 0) and (6, 6) is positive, let’s consider the coordinates of the points.

1. Identify the Coordinates: The first point is (0, 0) and the second point is (6, 6).

2. Understanding the Slope: The slope \( m \) of a line is calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

3. Substituting the Points: In our case:

   • \( (x_1, y_1) = (0, 0) \)
   • \( (x_2, y_2) = (6, 6) \)

   Substituting these values into the slope formula gives: \[ m = \frac{6 - 0}{6 - 0} = \frac{6}{6} = 1 \]

4. Conclusion About the Slope: Since the slope \( m = 1 \) is positive, we can conclude that:

   • The line rises as it moves from left to right on the coordinate plane.
   • Both the x-coordinate and y-coordinate increase as we move from the point (0, 0) to (6, 6).

Therefore, the slope is positive because as the x-value increases, the y-value also increases, indicating a direct relationship between the two variables.