Question

Find the slope between: (4,5) (3,5)

Answer 1

Answer:

If a line goes through both points, the slope of that line would be $0$.

Step-by-step explanation:

The slope of a line is equal to the ratio between the vertical change of this line "rise" and the corresponding horizontal change "run":

$\begin{aligned}\text{slope} &= \frac{\text{rise}}{\text{run}}\end{aligned}$.

If a non-vertical line goes through two distinct points $(x_{0},\, y_{0})$ and $(x_{1},\, y_{1})$ where $x_{0} \ne x_{1}$, the vertical change (rise) would be $(y_{1} - y_{0})$. The corresponding horizontal change would be $(x_{1} - x_{0})$. Thus, the slope of that line would be:

$\begin{aligned}\text{slope} &= \frac{\text{rise}}{\text{run}} \\ &= \frac{y_{1} - y_{0}}{x_{1} - x_{0}}\end{aligned}$.

In this question, $x_{0} = 4$ and $y_{0} = 5$ for the first point, whereas $x_{1} = 3$ and $y_{1} = 5$ for the second point.

The rise (vertical change) of this line from the first point to the second would be $(y_{1} - y_{0}) = 5 - 5 = 0$. The corresponding run (horizontal change) would be $(x_{1} - x_{0}) = 3 - 4 = -1$.

Thus, the slope of this line would be:

$\begin{aligned}\text{slope} &= \frac{\text{rise}}{\text{run}} \\ &= \frac{y_{1} - y_{0}}{x_{1} - x_{0}} \\ &= \frac{5 - 5}{3 - 4} \\&= \frac{0}{-1} \\ &= 0\end{aligned}$.

In other words, the slope of this line is $0$. Geometrically, a slope with a slope of $0\!$ would be horizontal. The vertical coordinate of all points on such line would be the same ($5$ for points on the line in this question.)