To determine the slope of the line passing through the points [tex]\((-5, 9)\)[/tex] and [tex]\((-5, 6)\)[/tex], we use the slope formula, which is given by:
[tex]\[
m = \frac{{y_2 - y_1}}{{x_2 - x_1}}
\][/tex]
Here, our coordinates are:
[tex]\[
(x_1, y_1) = (-5, 9)
\][/tex]
[tex]\[
(x_2, y_2) = (-5, 6)
\][/tex]
Plugging these coordinates into the slope formula, we get:
[tex]\[
m = \frac{{6 - 9}}{{-5 - (-5)}}
\][/tex]
First, calculate the differences in the coordinates:
- The difference in the [tex]\(y\)[/tex]-coordinates is:
[tex]\[
6 - 9 = -3
\][/tex]
- The difference in the [tex]\(x\)[/tex]-coordinates is:
[tex]\[
-5 - (-5) = -5 + 5 = 0
\][/tex]
Now, substitute these differences back into the slope formula:
[tex]\[
m = \frac{{-3}}{{0}}
\][/tex]
Since the denominator is zero, the fraction is undefined. In the context of geometry, this indicates that the line is vertical. Vertical lines have undefined slopes.
Therefore, the slope of the line passing through the points [tex]\((-5, 9)\)[/tex] and [tex]\((-5, 6)\)[/tex] is undefined, represented as infinity ([tex]\(\infty\)[/tex]).
