To determine the equation of the line that passes through the points (-6, 0) and (0, -4), we first need to find the slope of the line.
The formula for the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the points \((-6, 0)\) and \((0, -4)\):
\[ m = \frac{-4 - 0}{0 - (-6)} = \frac{-4}{0 + 6} = \frac{-4}{6} = -\frac{2}{3} \]
Now that we have the slope \( m = -\frac{2}{3} \), we can use the point-slope form of the line equation:
\[ y - y_1 = m(x - x_1) \]
We can choose one of the two points; let's use the point (0, -4):
\[ y - (-4) = -\frac{2}{3}(x - 0) \]
This simplifies to:
\[ y + 4 = -\frac{2}{3}x \]
Subtracting 4 from both sides gives:
\[ y = -\frac{2}{3}x - 4 \]
Thus, the correct equation of the line is:
\[ y = -\frac{2}{3}x - 4 \]
Now, let's match this with the options given:
- \(y = -\frac{2}{3}(x - 6)\) -- This would not correctly represent the line.
- \(y = -\frac{3}{2}(x - 6)\) -- This has the wrong slope.
- \(y = -\frac{2}{3}(x - 4)\) -- This represents a different line.
- \(y = -\frac{3}{2}(x - 4)\) -- This also has the wrong slope.
None of the provided options correctly represent the equation derived.
If you would like to use either point in the point-slope form again or check the options in different terms, or if you can provide more details about the adaptation of these options, I can help further.