To find the equation of a line that passes through the points [tex]\((-1, -1)\)[/tex] and [tex]\(2, -3\)[/tex], we proceed in the following steps:
1. Calculate the Slope (m):
The formula for the slope [tex]\(m\)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substituting in our points [tex]\((-1, -1)\)[/tex] and [tex]\((2, -3)\)[/tex]:
[tex]\[
m = \frac{-3 - (-1)}{2 - (-1)} = \frac{-3 + 1}{2 + 1} = \frac{-2}{3} = -\frac{2}{3}
\][/tex]
2. Calculate the y-intercept (b):
Use the point-slope form of the equation of a line [tex]\(y = mx + b\)[/tex]. We can use the slope we just calculated and one of the points, say [tex]\((-1, -1)\)[/tex], to find [tex]\(b\)[/tex].
Substitute [tex]\(m\)[/tex], [tex]\(x\)[/tex], and [tex]\(y\)[/tex]:
[tex]\[
-1 = -\frac{2}{3}(-1) + b
\][/tex]
Simplifying:
[tex]\[
-1 = \frac{2}{3} + b
\][/tex]
Solving for [tex]\(b\)[/tex]:
[tex]\[
b = -1 - \frac{2}{3} = -\frac{3}{3} - \frac{2}{3} = -\frac{5}{3}
\][/tex]
3. Form the Equation:
Using the slope [tex]\(m\)[/tex] and y-intercept [tex]\(b\)[/tex], the equation of the line in slope-intercept form [tex]\(y = mx + b\)[/tex] is:
[tex]\[
y = -\frac{2}{3}x - \frac{5}{3}
\][/tex]
4. Convert to Standard Form (Ax + By = C):
To convert the slope-intercept form to the standard form [tex]\(Ax + By = C\)[/tex], we need to get rid of the fractions. We can multiply every term by 3 to clear the denominators:
[tex]\[
3y = -2x - 5
\][/tex]
Rearrange the terms to get [tex]\(Ax + By = C\)[/tex]:
[tex]\[
2x + 3y = -5
\][/tex]
Looking at the given options, we can see that the result matches option B.
Thus, the correct equation is:
[tex]\[
2x + 3y = -5
\][/tex]
So, the correct answer is option B.
