To determine the equation of the line in point-slope form using the given point [tex]\((-2, -6)\)[/tex], we follow several steps:
1. Identify points to use:
We will use the points [tex]\((-4, -11)\)[/tex] and [tex]\((-2, -6)\)[/tex] to calculate the slope [tex]\(m\)[/tex].
2. 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:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Plug in the given points:
[tex]\[
m = \frac{-6 - (-11)}{-2 - (-4)} = \frac{-6 + 11}{-2 + 4} = \frac{5}{2}
\][/tex]
3. Write the point-slope form:
The point-slope form of a linear equation is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Using the point [tex]\((-2, -6)\)[/tex] [tex]\((x_1, y_1)\)[/tex] and the calculated slope [tex]\(m = \frac{5}{2}\)[/tex]:
[tex]\[
y - (-6) = \frac{5}{2}(x - (-2))
\][/tex]
Simplify the expression:
[tex]\[
y + 6 = \frac{5}{2}(x + 2)
\][/tex]
4. Compare with given options:
- [tex]\(y - 6 = \frac{5}{2}(x - 2)\)[/tex]
- [tex]\(y - 6 = \frac{2}{5}(x - 2)\)[/tex]
- [tex]\(y + 8 = \frac{2}{5}(x + 2)\)[/tex]
Clearly, [tex]\(y + 6 = \frac{5}{2}(x + 2)\)[/tex] matches our derived equation.
Thus, the correct equation is:
[tex]\[
y + 6 = \frac{5}{2}(x + 2)
\][/tex]
