Verified Answer :

  • Identify the coordinates of the two points: \((x_1, y_1) = (1, -9)\) and \((x_2, y_2) = (4, -7)\).
  • Apply the slope formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
  • Substitute the values into the formula to get \(m = \frac{-7 - (-9)}{4 - 1}\).
  • Simplify the expression to find the final slope: \(\boxed{\frac{2}{3}}\).

Explanation

  1. Understanding the Slope Formula

To find the slope of the line passing through the points \( (1, -9) \) and \( (4, -7) \), we need to determine the rate of change between these two points. The slope, denoted by \( m \), represents the vertical change (rise) divided by the horizontal change (run) between any two distinct points on a line. The formula for the slope \( m \) given two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
In this case, our points are \( (x_1, y_1) = (1, -9) \) and \( (x_2, y_2) = (4, -7) \).

  1. Calculating the Slope

Now, we substitute the coordinates of our points into the slope formula:
\[m = \frac{-7 - (-9)}{4 - 1}\]
Next, we perform the arithmetic operations in the numerator and the denominator:

  • Numerator: \(-7 - (-9) = -7 + 9 = 2\)
  • Denominator: \(4 - 1 = 3\)

So, the slope is:
\[m = \frac{2}{3}\]
This fraction is already in its simplest form.

  1. Conclusion

The slope of the line passing through the points \( (1, -9) \) and \( (4, -7) \) is \(\boxed{\frac{2}{3}}\).

Examples

In daily life, the concept of slope is essential in various fields, such as engineering and construction. For example, when building a ramp for accessibility, engineers calculate the slope to ensure it is not too steep for wheelchair users. If a ramp rises 2 inches for every 3 inches of horizontal distance, the slope is 2/3. This ensures safety and comfort, demonstrating how calculating the rate of change helps us design functional and accessible structures.

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