(a) Find the slope of the line through [tex]$(-2,-8)$[/tex] and [tex]$(-7,-58)$[/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.

(a) Select the correct choice below and, if necessary, fill in the answer box to complete your choice.



Answer :

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.