What is the slope of the line that passes through the points)(7,−4) and (7, 0) ? Write your answer in simplest form.
The Correct Answer and Explanation is:
The problem asks for the slope of the line that passes through the points ( (7, -4) ) and ( (7, 0) ). To find the slope, we can use the slope formula:
[
m = \frac{y_2 – y_1}{x_2 – x_1}
]
Where ( m ) represents the slope of the line, and ( (x_1, y_1) ) and ( (x_2, y_2) ) are the coordinates of two points on the line.
In this case, the two points are:
[
(x_1, y_1) = (7, -4)
]
[
(x_2, y_2) = (7, 0)
]
Now, substitute these values into the slope formula:
[
m = \frac{0 – (-4)}{7 – 7}
]
[
m = \frac{0 + 4}{0}
]
[
m = \frac{4}{0}
]
Here, we encounter a problem: division by zero. The denominator is 0, meaning the slope is undefined. This indicates that the line is vertical.
Explanation:
The slope of a line represents the “rise” (change in ( y )-values) over the “run” (change in ( x )-values). In this case, the ( x )-coordinates of both points are the same (( x_1 = x_2 = 7 )). This means that the line does not change horizontally; it is vertical.
For vertical lines, the slope is always undefined because there is no horizontal change (( \Delta x = 0 )), and dividing by zero is not possible mathematically. Therefore, the slope of the line passing through the points ( (7, -4) ) and ( (7, 0) ) is undefined.
To summarize:
- The slope formula gives a result of ( \frac{4}{0} ), which is undefined.
- This means the line is vertical, and vertical lines do not have a defined slope.