To find the equation of a line passing through the point [tex]\((6, -16)\)[/tex] with a slope of [tex]\(-\frac{2}{3}\)[/tex], we can use the point-slope form of a linear equation. 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 of the line.
Here, we have the point [tex]\((6, -16)\)[/tex] and the slope [tex]\(m = -\frac{2}{3}\)[/tex]. Substituting these values into the point-slope form, we get:
[tex]\[ y - (-16) = -\frac{2}{3}(x - 6). \][/tex]
Simplifying, this becomes:
[tex]\[ y + 16 = -\frac{2}{3}(x - 6). \][/tex]
So, the equation representing the line passing through the point [tex]\((6, -16)\)[/tex] with a slope of [tex]\(-\frac{2}{3}\)[/tex] is:
[tex]\[ y + 16 = -\frac{2}{3}(x - 6). \][/tex]
Therefore, the correct option is:
[tex]\[ y + 16 = -\frac{2}{3}(x - 6). \][/tex]
Hence, the answer is option 4.
