Question

Find the equation of a line that passes through the points (2,5) and (2,12)

Answer 1

Determining the slope of the line:

We know the following:

$\text{Slope formula} = \dfrac{(y_2 - y_1)}{(x_2 - x_1)}$

Note: Slopes can only be determined if the lines are not vertical.

Since the x-coordinate of the points that pass through the line are the same, when the points are substituted in the slope formula, the slope will result in "undefined" (Work shown below).

$\text{Slope formula} = \dfrac{12 - 5}{2 - 2}$

$\text{Slope formula} = \dfrac{7}{0} = \text{unde}\text{fined} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [\text{Any non-number divided by 0 is unde}\text{fined}]$

Now, let's see if the vertical line will intersect the x-axis or a y-axis.

Determining if the line will intersect the x or y-axis

To determine if the vertical line will intersect the x or y-axis, we must look for the intersection of the line on the x or y-axis. The vertical line will be parallel to the y-axis and will intersect the x-axis.

Therefore, the line will intersect the x-axis.

Replacing values in an equation:

Since the line does not intersect the y-axis, the equation of the line on the left-hand side will have an x-variable.

⇒ x = ?

The x-intercept will be substituted on the right-hand side of the equation.  Since the line that passes through the two points [(2,5) and (2,12)] have the "2" as their x-coordinates, the x-intercept will also be 2. Therefore,

⇒ x = 2