Sure, let's solve the problem step-by-step:
### (a) Find the slope of the line through [tex]\((-2, -8)\)[/tex] and [tex]\((-7, -58)\)[/tex]:
To find the slope of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex], we use the formula:
[tex]\[ \text{slope} (m) = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substitute the given points into the formula:
[tex]\[ (x_1, y_1) = (-2, -8) \][/tex]
[tex]\[ (x_2, y_2) = (-7, -58) \][/tex]
[tex]\[ m = \frac{-58 - (-8)}{-7 - (-2)} \][/tex]
[tex]\[ m = \frac{-58 + 8}{-7 + 2} \][/tex]
[tex]\[ m = \frac{-50}{-5} \][/tex]
[tex]\[ m = 10 \][/tex]
So, the slope of the line through [tex]\((-2, -8)\)[/tex] and [tex]\((-7, -58)\)[/tex] is [tex]\(10.0\)[/tex].
### (b) Based on the slope, indicate whether the line through the points rises from left to right, falls from left to right, is horizontal, or is vertical:
The slope of the line we calculated is [tex]\(10.0\)[/tex].
- If the slope is positive ([tex]\(m > 0\)[/tex]), the line rises from left to right.
- If the slope is negative ([tex]\(m < 0\)[/tex]), the line falls from left to right.
- If the slope is zero ([tex]\(m = 0\)[/tex]), the line is horizontal.
- If the denominator in the slope formula is zero, this would indicate a vertical line (undefined slope, but not applicable here).
Since the slope [tex]\(10.0\)[/tex] is positive:
The line rises from left to right.
### Solution:
(a) The slope of the line through [tex]\((-2, -8)\)[/tex] and [tex]\((-7, -58)\)[/tex] is [tex]\(10.0\)[/tex].
(b) The line rises from left to right.
