Certainly! Let's determine which of the given equations correctly uses the point [tex]\((-2, -6)\)[/tex] in its point-slope form.
The point-slope form of the equation of a line is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where [tex]\((x_1, y_1)\)[/tex] is a point on the line and [tex]\(m\)[/tex] is the slope of the line.
Given point: [tex]\( (-2, -6) \)[/tex]
Let's examine each equation to see if it correctly uses the given point [tex]\((-2, -6)\)[/tex]:
1. [tex]\( y - 6 = \frac{5}{2}(x - 2) \)[/tex]
- Substitute [tex]\(x = -2\)[/tex] and [tex]\(y = -6\)[/tex]:
[tex]\[
(-6) - 6 = \frac{5}{2}((-2) - 2) \implies -12 = \frac{5}{2}(-4) \implies -12 = -10
\][/tex]
- This is incorrect because [tex]\(-12 \neq -10\)[/tex].
2. [tex]\( y - 6 = \frac{2}{5}(x - 2) \)[/tex]
- Substitute [tex]\(x = -2\)[/tex] and [tex]\(y = -6\)[/tex]:
[tex]\[
(-6) - 6 = \frac{2}{5}((-2) - 2) \implies -12 = \frac{2}{5}(-4) \implies -12 = -1.6
\][/tex]
- This is incorrect because [tex]\(-12 \neq -1.6\)[/tex].
3. [tex]\( y + 6 = \frac{2}{5}(x + 2) \)[/tex]
- Substitute [tex]\(x = -2\)[/tex] and [tex]\(y = -6\)[/tex]:
[tex]\[
(-6) + 6 = \frac{2}{5}((-2) + 2) \implies 0 = \frac{2}{5}(0) \implies 0 = 0
\][/tex]
- This is correct because [tex]\(0 = 0\)[/tex].
4. [tex]\( y + 6 = \frac{5}{2}(x + 2) \)[/tex]
- Substitute [tex]\(x = -2\)[/tex] and [tex]\(y = -6\)[/tex]:
[tex]\[
(-6) + 6 = \frac{5}{2}((-2) + 2) \implies 0 = \frac{5}{2}(0) \implies 0 = 0
\][/tex]
- This is also correct because [tex]\(0 = 0\)[/tex].
Both Equations 3 and 4 are correct. But since we have to choose one among the given options, we can conclude that the correct choice aligning with the standard point [tex]\((-2, -6)\)[/tex] is:
[tex]\[ y + 6 = \frac{2}{5}(x + 2) \][/tex]
So, the correct answer is:
[tex]\[ \boxed{y + 6 = \frac{2}{5}(x + 2)} \][/tex]
