What is the slope of the line that passes through the points (-1, 1) and (-1, 4)? Write your answer in simplest form.
The problem asks for the slope of the line passing through two points: \((-1, 1)\) and \((-1, 4)\). Let these points be \((x_1, y_1) = (-1, 1)\) and \((x_2, y_2) = (-1, 4)\). A line's slope, denoted by \(m\), is defined as the change in the vertical direction divided by the change in the horizontal direction, which can be expressed as:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
This formula is valid as long as \(x_2
eq x_1\).
We substitute the given points \((-1, 1)\) and \((-1, 4)\) into the slope formula:
\[m = \frac{4 - 1}{-1 - (-1)}\]
Calculating the numerator, we have \(4 - 1 = 3\).
Calculating the denominator, we have \(-1 - (-1) = -1 + 1 = 0\).
So, the slope is:
\[m = \frac{3}{0}\]
Division by zero is undefined in mathematics. Since the x-coordinates of the two points are the same, the line passing through them is a vertical line. A vertical line does not have a defined slope because the run is zero.
A vertical line is like a perfectly plumb wall in a house or a sheer cliff face. In engineering and construction, knowing that a vertical line has an "undefined" slope is crucial, as it tells you the line is perfectly perpendicular to the ground and doesn't "tilt" in either direction, which would require a specific numerical slope calculation. This concept helps in accurately surveying land and ensuring structural stability in architecture.