Sure, let's determine which equation represents the line passing through the point [tex]\((-2, -3)\)[/tex] with a slope of -6.
We use the point-slope form of a line equation first:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Where:
- [tex]\((x_1, y_1)\)[/tex] is the point on the line, which in this case is [tex]\((-2, -3)\)[/tex].
- [tex]\(m\)[/tex] is the slope, which here is [tex]\(-6\)[/tex].
Substitute these values into the point-slope form equation:
[tex]\[ y - (-3) = -6(x - (-2)) \][/tex]
[tex]\[ y + 3 = -6(x + 2) \][/tex]
Next, we simplify and convert this equation into the slope-intercept form ([tex]\(y = mx + b\)[/tex]):
[tex]\[ y + 3 = -6x - 12 \][/tex]
Subtract 3 from both sides to isolate [tex]\(y\)[/tex]:
[tex]\[ y = -6x - 12 - 3 \][/tex]
[tex]\[ y = -6x - 15 \][/tex]
So, the equation that represents the line passing through the point [tex]\((-2, -3)\)[/tex] with a slope of [tex]\(-6\)[/tex] is:
[tex]\[ y = -6x - 15 \][/tex]
Therefore, the correct choice is:
[tex]\[ y = -6x - 15 \][/tex]
