- Identify the two points \((15, 29)\) and \((57, -11)\).
- Apply the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\) to compute the rise over run: \(\frac{-11 - 29}{57 - 15} = \frac{-40}{42}\).
- Simplify the resulting fraction by dividing both the numerator and the denominator by their greatest common divisor, 2.
- The final simplified slope is \(\boxed{-\frac{20}{21}}\).
Explanation
- Identifying the Formula and Points
To find the slope of the line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\), we use the slope formula:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
In this problem, the given points are \((x_1, y_1) = (15, 29)\) and \((x_2, y_2) = (57, -11)\). By identifying these coordinates, we can proceed to substitute them into the slope formula to determine the slope \(m\).
- Calculating the Slope Expression
Substitute the identified values into the slope formula:
\[m = \frac{-11 - 29}{57 - 15}\]
First, calculate the numerator:
\[-11 - 29 = -40\]
Next, calculate the denominator:
\[57 - 15 = 42\]
Thus, the slope is given by the fraction:
\[m = \frac{-40}{42}\]
- Simplifying the Fraction
To simplify the fraction \(\frac{-40}{42}\), we find the greatest common divisor (GCD) of 40 and 42. Since both numbers are even, we can divide both by 2:
\[-40 \div 2 = -20\]
\[42 \div 2 = 21\]
The fraction \(\frac{-20}{21}\) cannot be simplified further, as 20 and 21 have no common factors other than 1. Therefore, the simplified slope of the line is \(-\frac{20}{21}\).
Examples
Calculating the slope is fundamental in many real-life applications. For instance, if you are designing a wheelchair ramp, the slope determines how steep or accessible the ramp will be. A slope of \(\frac{\text{rise}}{\text{run}}\) ensures that the rise of the ramp (height) relative to its run (horizontal distance) complies with safety standards, typically requiring a slope no greater than 1:12. By applying the slope formula \(\frac{y_2 - y_1}{x_2 - x_1}\), engineers and builders can precisely determine the incline of structures like roofs, roads, and staircases.