To determine the slope of a line that passes through two points [tex]\((-1, \frac{1}{3})\)[/tex] and [tex]\( (0, -\frac{1}{3}) \)[/tex], we use the formula for the slope [tex]\((m)\)[/tex] of a line passing through points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex]:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substitute the given coordinates into the formula:
[tex]\[
x_1 = -1, \quad y_1 = \frac{1}{3}
\][/tex]
[tex]\[
x_2 = 0, \quad y_2 = -\frac{1}{3}
\][/tex]
Now, plug these values into the slope formula:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-\frac{1}{3} - \frac{1}{3}}{0 - (-1)}
\][/tex]
Simplify the differences in the numerator and the denominator:
[tex]\[
m = \frac{-\frac{1}{3} - \frac{1}{3}}{0 + 1} = \frac{-\frac{1}{3} - \frac{1}{3}}{1} = \frac{-\frac{2}{3}}{1}
\][/tex]
Since dividing by 1 does not change the value, we have:
[tex]\[
m = -\frac{2}{3}
\][/tex]
Hence, the slope of the line passing through the points [tex]\((-1, \frac{1}{3})\)[/tex] and [tex]\( (0, -\frac{1}{3})\)[/tex] is [tex]\(-\frac{2}{3}\)[/tex]. Therefore, the correct answer is:
(A) [tex]\(-\frac{2}{3}\)[/tex]
