Question

What is the slope of the line $x=-2?$

Answer 1

We know that the slope of a line determines the steepness of the line. If $\left( {{x}_{1}},{{y}_{1}} \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)$ are two points through which the line of discussion passes then the slope of the line can be found by $\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}.$

Complete step by step answer:
Let us consider the given line $x=-2.$
We are asked to find the slope of the given line.
We can clearly say that the given line is a vertical line that makes an angle of $90{}^\circ$ with the $x-$axis.
We can say that this line makes the $x-$ coordinates of the points fixed for the line is vertical and the value is $x=-2.$
We learn that whatever be the value of $y-$ coordinate, the value of $x-$coordinate remains the same.
Let us suppose that $\left( {{x}_{1}},{{y}_{1}} \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)$ are two points through which the given line passes.
Then, we can write the points as $\left( -2,{{y}_{1}} \right)$ and $\left( -2,{{y}_{2}} \right).$
We know that the slope of a line can be found by using the formula given by $\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}.$
Let us substitute the values and find what we obtain.
So, we will get $\dfrac{{{y}_{2}}-{{y}_{1}}}{-2+2}.$
From this, we will get $\dfrac{{{y}_{2}}-{{y}_{1}}}{0}.$
So, we have obtained a fraction with the denominator equal to zero.
We know that when the denominator is zero, the fraction or quotient is said to be undefined.

Hence, we can conclude that the slope of the given vertical line is undefined.

Note: From what we have obtained, we can generalize the fact that the slope of the given vertical line is undefined. So, the generalization will lead us to the fact that the slope of any vertical line is undefined. On the other hand, we can find that the slopes of all horizontal lines are equal to zero for the points that have the same $y-$coordinates.