To determine the slope of a line passing through two points, we use the slope formula:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Here, we have two points: [tex]\((-9, -2)\)[/tex] and [tex]\((-9, -6)\)[/tex].
Let:
[tex]\[ (x_1, y_1) = (-9, -2) \][/tex]
[tex]\[ (x_2, y_2) = (-9, -6) \][/tex]
Substitute these values into the slope formula:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-6 - (-2)}{-9 - (-9)} \][/tex]
[tex]\[ m = \frac{-6 + 2}{-9 + 9} \][/tex]
[tex]\[ m = \frac{-4}{0} \][/tex]
We encounter a problem here. The denominator is zero, which means the fraction is undefined. This occurs because the x-coordinates of both points are the same ([tex]\(x_1 = x_2 = -9\)[/tex]). In such cases, the line is vertical.
A vertical line has an undefined slope, often referred to as infinity.
Thus, the slope of the line that passes through the points [tex]\((-9, -2)\)[/tex] and [tex]\((-9, -6)\)[/tex] is:
[tex]\[ \text{undefined (infinity)} \][/tex]
