To find the equation of the line passing through the points [tex]\((3, -7)\)[/tex] and [tex]\((-6, -13)\)[/tex], we need to determine the slope [tex]\(m\)[/tex] and the y-intercept [tex]\(b\)[/tex] of the line using the slope-intercept form [tex]\(y = mx + b\)[/tex].
1. Calculate the Slope [tex]\(m\)[/tex]:
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]
Plugging in the coordinates of the given points [tex]\((3, -7)\)[/tex] and [tex]\((-6, -13)\)[/tex], we get:
[tex]\[
m = \frac{-13 - (-7)}{-6 - 3} = \frac{-13 + 7}{-6 - 3} = \frac{-6}{-9} = \frac{2}{3}
\][/tex]
2. Determine the Y-Intercept [tex]\(b\)[/tex]:
The y-intercept [tex]\(b\)[/tex] can be found using the slope-intercept form [tex]\(y = mx + b\)[/tex]. We select one of the given points, say [tex]\((3, -7)\)[/tex], and substitute [tex]\(x = 3\)[/tex] and [tex]\(y = -7\)[/tex] into the equation along with the slope [tex]\(m = \frac{2}{3}\)[/tex]:
[tex]\[
-7 = \left(\frac{2}{3}\right)(3) + b
\][/tex]
Simplifying the equation:
[tex]\[
-7 = 2 + b
\][/tex]
Solving for [tex]\(b\)[/tex]:
[tex]\[
b = -7 - 2 = -9
\][/tex]
3. Write the Equation of the Line:
Now that we have the slope [tex]\(m = \frac{2}{3}\)[/tex] and the y-intercept [tex]\(b = -9\)[/tex], we can write the equation of the line in slope-intercept form:
[tex]\[
y = \frac{2}{3}x - 9
\][/tex]
Therefore, the equation of the line passing through the points [tex]\((3, -7)\)[/tex] and [tex]\((-6, -13)\)[/tex] is:
[tex]\[
y = \frac{2}{3}x - 9
\][/tex]
