Verified Answer :

  • Identify the coordinates of the two points as \((-9, -7)\) and \((-11, -6)\).
  • Use the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
  • Calculate the difference in y-coordinates (\(1\)) and x-coordinates (\(-2\)).
  • The slope of the line is \(\boxed{-\frac{1}{2}}\).

Explanation

  1. Understanding the Slope Formula

The problem requires calculating the slope of a line passing through two points: \(P_1 = (-9, -7)\) and \(P_2 = (-11, -6)\). The slope \(m\) of a line passing through points \((x_1, y_1)\) and \((x_2, y_2)\) is defined by the change in the vertical direction divided by the change in the horizontal direction:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
provided \(x_1 eq x_2\). We will use this formula to find the slope by substituting the given coordinates.

  1. Applying the Coordinates to the Formula

Given the points \((x_1, y_1) = (-9, -7)\) and \((x_2, y_2) = (-11, -6)\), we identify:

  • \(x_1 = -9\)
  • \(y_1 = -7\)
  • \(x_2 = -11\)
  • \(y_2 = -6\)

Substituting these values into the slope formula, we get:
\[m = \frac{-6 - (-7)}{-11 - (-9)}\]

  1. Calculating and Simplifying the Slope

Now we simplify the numerator and denominator:

  • The numerator: \(-6 - (-7) = -6 + 7 = 1\)
  • The denominator: \(-11 - (-9) = -11 + 9 = -2\)

So, the slope \(m\) is:
\[m = \frac{1}{-2} = -\frac{1}{2}\]
The slope of the line is \(-1/2\).

Examples

In real-world applications, the concept of a slope represents the rate of change of one variable with respect to another. For example, if you are analyzing the change in elevation of a hiking trail over a certain distance, the slope helps you understand how steep that section of the trail is. If the trail drops 1 unit in elevation for every 2 units of horizontal distance, the slope of that section is -1/2, indicating a downward incline. This mathematical tool is essential in fields like engineering, economics, and geography to track trends and gradients effectively.

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