To find the equation of the line passing through the points [tex]\((-5, -2)\)[/tex] and [tex]\((3, -1)\)[/tex], we need to determine two key components of the linear equation [tex]\(y = mx + b\)[/tex]: the slope [tex]\(m\)[/tex] and the y-intercept [tex]\(b\)[/tex].
1. Calculate the Slope [tex]\(m\)[/tex]:
The slope [tex]\(m\)[/tex] is given by the formula:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substituting in the points [tex]\((-5, -2)\)[/tex] and [tex]\((3, -1)\)[/tex]:
[tex]\[
m = \frac{-1 - (-2)}{3 - (-5)} = \frac{-1 + 2}{3 + 5} = \frac{1}{8}
\][/tex]
2. Calculate the Y-Intercept [tex]\(b\)[/tex]:
Using the slope [tex]\(m = \frac{1}{8}\)[/tex] and one of the given points, say [tex]\((-5, -2)\)[/tex], we can find the y-intercept [tex]\(b\)[/tex]. The equation for the y-intercept is derived from the point-slope form of the equation:
[tex]\[
y = mx + b
\][/tex]
Substituting [tex]\(x = -5\)[/tex], [tex]\(y = -2\)[/tex], and [tex]\(m = \frac{1}{8}\)[/tex] into the equation:
[tex]\[
-2 = \frac{1}{8}(-5) + b
\][/tex]
Solving for [tex]\(b\)[/tex]:
[tex]\[
-2 = -\frac{5}{8} + b
\][/tex]
[tex]\[
b = -2 + \frac{5}{8}
\][/tex]
Converting [tex]\(-2\)[/tex] to a fraction with a common denominator of 8:
[tex]\[
-2 = -\frac{16}{8}
\][/tex]
Adding the fractions:
[tex]\[
b = -\frac{16}{8} + \frac{5}{8} = -\frac{11}{8}
\][/tex]
3. Form the Equation of the Line:
Now we have both the slope [tex]\(m = \frac{1}{8}\)[/tex] and the y-intercept [tex]\(b = -\frac{11}{8}\)[/tex]. Substituting these values into the slope-intercept form [tex]\(y = mx + b\)[/tex], we get:
[tex]\[
y = \frac{1}{8}x - \frac{11}{8}
\][/tex]
So, the correct equation of the line that passes through the points [tex]\((-5, -2)\)[/tex] and [tex]\((3, -1)\)[/tex] is:
[tex]\[
\boxed{y = \frac{1}{8}x - \frac{11}{8}}
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{b}
\][/tex]
