To determine which equation represents the line that passes through the point [tex]\((-9, -3)\)[/tex] and has a slope of [tex]\(-6\)[/tex], we can use the point-slope form of the equation of a line. The point-slope form is given by:
[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.
Given:
[tex]\( (x_1, y_1) = (-9, -3) \)[/tex]
[tex]\( m = -6 \)[/tex]
Plugging these values into the point-slope form, we get:
[tex]\[ y - (-3) = -6(x - (-9)) \][/tex]
This simplifies to:
[tex]\[ y + 3 = -6(x + 9) \][/tex]
So, the equation that represents the line passing through the point [tex]\((-9, -3)\)[/tex] with a slope of [tex]\(-6\)[/tex] is:
[tex]\[ y + 3 = -6(x + 9) \][/tex]
Among the given options, this corresponds to:
[tex]\[ y + 3 = -6(x + 9) \][/tex]
Thus, the correct equation is:
[tex]\[ y + 3 = -6(x + 9) \][/tex]
In the given multiple-choice format, the correct answer is the fourth option:
[tex]\[ y+3=-6(x+9) \][/tex]
