Answer:
If a line goes through both points, the slope of that line would be [tex]0[/tex].
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":
[tex]\begin{aligned}\text{slope} &= \frac{\text{rise}}{\text{run}}\end{aligned}[/tex].
If a non-vertical line goes through two distinct points [tex](x_{0},\, y_{0})[/tex] and [tex](x_{1},\, y_{1})[/tex] where [tex]x_{0} \ne x_{1}[/tex], the vertical change (rise) would be [tex](y_{1} - y_{0})[/tex]. The corresponding horizontal change would be [tex](x_{1} - x_{0})[/tex]. Thus, the slope of that line would be:
[tex]\begin{aligned}\text{slope} &= \frac{\text{rise}}{\text{run}} \\ &= \frac{y_{1} - y_{0}}{x_{1} - x_{0}}\end{aligned}[/tex].
In this question, [tex]x_{0} = 4[/tex] and [tex]y_{0} = 5[/tex] for the first point, whereas [tex]x_{1} = 3[/tex] and [tex]y_{1} = 5[/tex] for the second point.
The rise (vertical change) of this line from the first point to the second would be [tex](y_{1} - y_{0}) = 5 - 5 = 0[/tex]. The corresponding run (horizontal change) would be [tex](x_{1} - x_{0}) = 3 - 4 = -1[/tex].
Thus, the slope of this line would be:
[tex]\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}[/tex].
In other words, the slope of this line is [tex]0[/tex]. Geometrically, a slope with a slope of [tex]0\![/tex] would be horizontal. The vertical coordinate of all points on such line would be the same ([tex]5[/tex] for points on the line in this question.)
