What is the slope of the line that passes through the points (1, -6) and (-2, -8)? Write your answer in simplest form.
To find the slope of the line passing through two points, we use the coordinates of these points: \(P_1 = (x_1, y_1) = (1, -6)\) and \(P_2 = (x_2, y_2) = (-2, -8)\). The slope \(m\) of a line is defined as the change in the vertical direction (rise) divided by the change in the horizontal direction (run) between two points on the line. The mathematical formula for this is:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
This formula allows us to calculate how steeply the line rises or falls as we move from the first point to the second point.
Now, we substitute the coordinates of our given points \((1, -6)\) and \((-2, -8)\) into the slope formula:
\[m = \frac{-8 - (-6)}{-2 - 1}\]
First, we simplify the numerator, which involves subtracting a negative number:
\(-8 - (-6) = -8 + 6 = -2\)
Next, we simplify the denominator:
\(-2 - 1 = -3\)
So, the slope \(m\) is the fraction formed by these two results:
\[m = \frac{-2}{-3}\]
Since dividing a negative number by another negative number results in a positive value, we have:
\[m = \frac{2}{3}\]
The slope of the line is \(\frac{2}{3}\).
The slope of the line passing through the points \((1, -6)\) and \((-2, -8)\) is determined to be \(\frac{2}{3}\). Thus, the final answer is \(\boxed{\frac{2}{3}}\).
In real life, the concept of slope is used to describe the steepness of objects like ramps, roads, or roofs. For example, if a wheelchair ramp rises \(2\) feet for every \(3\) feet of horizontal distance, we say the slope of the ramp is \(\frac{2}{3}\). This information is crucial for construction, safety regulations, and ensuring accessibility in building design. Understanding how to calculate slope allows engineers and architects to design structures that are both functional and safe.