To determine which of the given equations represents a line that passes through the points [tex]\((-6,1)\)[/tex] and [tex]\((0,-3)\)[/tex], we start by calculating the slope of the line passing through these points.
1. Calculate the slope [tex]\(m\)[/tex]:
The formula to calculate the slope 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]
Given the points [tex]\((-6,1)\)[/tex] and [tex]\((0,-3)\)[/tex]:
[tex]\[
m = \frac{-3 - 1}{0 + 6} = \frac{-4}{6} = -\frac{2}{3}
\][/tex]
2. Identify the line equation:
The general form of a line equation is:
[tex]\[
y = mx + b
\][/tex]
Substituting the slope [tex]\(m = -\frac{2}{3}\)[/tex] into the equation, we get:
[tex]\[
y = -\frac{2}{3}x + b
\][/tex]
To find the y-intercept [tex]\(b\)[/tex], we use one of the given points, [tex]\((-6,1)\)[/tex]:
[tex]\[
1 = -\frac{2}{3}(-6) + b
\][/tex]
Simplifying:
[tex]\[
1 = 4 + b
\][/tex]
[tex]\[
b = 1 - 4 = -3
\][/tex]
So, the equation of the line is:
[tex]\[
y = -\frac{2}{3}x - 3
\][/tex]
3. Check the given equations:
- Equation I: [tex]\(y = -\frac{2}{3}x - 2\)[/tex]
- Slope is [tex]\(-\frac{2}{3}\)[/tex], which matches the slope we found.
- Substituting [tex]\((-6, 1)\)[/tex]:
[tex]\[
1 = -\frac{2}{3}(-6) - 2
\][/tex]
[tex]\[
1 = 4 - 2 = 2
\][/tex]
This is incorrect because 1 does not equal 2. So Equation I does not pass through the point [tex]\((-6, 1)\)[/tex].
- Equation II: [tex]\(y + 9 = \frac{2}{3}(x - 9)\)[/tex]
- Transform to slope-intercept form:
[tex]\[
y = \frac{2}{3}x - 6 - 9
\][/tex]
[tex]\[
y = \frac{2}{3}x - 15
\][/tex]
- Slope is [tex]\(\frac{2}{3}\)[/tex], which does not match the slope we found. So Equation II does not match the slope of the line passing through the given points.
Based on these calculations, neither of the given equations passes through the points [tex]\((-6,1)\)[/tex] and [tex]\((0,-3)\)[/tex]. Therefore, the correct answer is:
Neither
