What is the slope of the line that passes through the points (-9, -9) and (-9, -7)? Write your answer in simplest form.
We are given two points: \(P_1 = (-9, -9)\) and \(P_2 = (-9, -7)\). To find the slope \(m\) of the line passing through these two points, we use the standard slope formula:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points. We can identify them as \(x_1 = -9\), \(y_1 = -9\), \(x_2 = -9\), and \(y_2 = -7\).
Now, we substitute the coordinates into the formula.
The change in \(y\) (the rise) is:
\[\Delta y = y_2 - y_1 = -7 - (-9) = -7 + 9 = 2\]
The change in \(x\) (the run) is:
\[\Delta x = x_2 - x_1 = -9 - (-9) = -9 + 9 = 0\]
Substituting these values into the slope formula, we get:
\[m = \frac{2}{0}\]
In mathematics, division by zero is undefined. This occurs whenever the x-coordinates of the two points are identical, meaning the line is vertical. A vertical line does not have a defined slope because there is no run, only rise. Therefore, the slope of the line passing through \((-9, -9)\) and \((-9, -7)\) is undefined.
Understanding the concept of an undefined slope is crucial in various fields, such as architecture and engineering. For example, if you are designing a ladder that is perfectly vertical against a wall, the line representing the ladder would have an undefined slope. This distinction helps engineers differentiate between surfaces that are slanted—which can be measured with a specific slope or grade—and those that are perfectly vertical.