What is the slope of the line that passes through the points (7, -5) and (16, 7)? Write your answer in simplest form.
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, which represents the ratio of the vertical change (rise) to the horizontal change (run). The coordinates of the given points are \((x_1, y_1) = (7, -5)\) and \((x_2, y_2) = (16, 7)\). The formula for the slope \(m\) is:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
Now, we substitute the given coordinate values into the slope formula:
\[m = \frac{7 - (-5)}{16 - 7}\]
First, calculate the numerator:
\[7 - (-5) = 7 + 5 = 12\]
Next, calculate the denominator:
\[16 - 7 = 9\]
Thus, the slope is:
\[m = \frac{12}{9}\]
To simplify the fraction \(\frac{12}{9}\), we divide both the numerator and the denominator by their greatest common divisor, which is \(3\):
\[m = \frac{12 \div 3}{9 \div 3} = \frac{4}{3}\]
Therefore, the slope of the line passing through the points \((7, -5)\) and \((16, 7)\) is \(\frac{4}{3}\). The final answer is \(\boxed{\frac{4}{3}}\).
The concept of slope is fundamental in fields like civil engineering and architecture. For example, when building a wheelchair ramp, the slope (or grade) must be carefully controlled to ensure safety and accessibility. A slope of \(\frac{4}{3}\) means that for every \(3\) units of horizontal distance, the ramp rises \(4\) units. Understanding how to calculate this ratio allows builders to design structures that are both functional and compliant with safety regulations.