To determine the slope of a line that passes through two points, we use the slope formula:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Here, the two points given are [tex]\( A(-3, -\frac{7}{2}) \)[/tex] and [tex]\( B(2, -4) \)[/tex].
First, let's identify the coordinates:
- Point [tex]\( A \)[/tex] has coordinates [tex]\( x_1 = -3 \)[/tex] and [tex]\( y_1 = -\frac{7}{2} \)[/tex].
- Point [tex]\( B \)[/tex] has coordinates [tex]\( x_2 = 2 \)[/tex] and [tex]\( y_2 = -4 \)[/tex].
Substitute these coordinates into the slope formula:
[tex]\[
m = \frac{-4 - \left(-\frac{7}{2}\right)}{2 - (-3)}
\][/tex]
Simplify inside the numerator and denominator:
- Subtract the y-coordinates in the numerator:
[tex]\[
-4 - \left(-\frac{7}{2}\right) = -4 + \frac{7}{2}
\][/tex]
To combine these terms, convert [tex]\(-4\)[/tex] to a fraction with a denominator of 2:
[tex]\[
-4 = -\frac{8}{2}
\][/tex]
So,
[tex]\[
-4 + \frac{7}{2} = -\frac{8}{2} + \frac{7}{2} = -\frac{8}{2} + \frac{7}{2} = -\frac{1}{2}
\][/tex]
- Subtract the x-coordinates in the denominator:
[tex]\[
2 - (-3) = 2 + 3 = 5
\][/tex]
Now substitute back into the formula:
[tex]\[
m = \frac{-\frac{1}{2}}{5}
\][/tex]
This simplifies to:
[tex]\[
m = -\frac{1}{2} \times \frac{1}{5} = -\frac{1}{10}
\][/tex]
Thus, the slope of the line passing through the points [tex]\(\left(-3, -\frac{7}{2}\right)\)[/tex] and [tex]\((2, -4)\)[/tex] is:
[tex]\[
-\frac{1}{10}
\][/tex]
Therefore, the correct answer is:
[tex]\[
-\frac{1}{10}
\][/tex]
