To write the point-slope form of the equation of a line that passes through the points [tex]\((-4, 7)\)[/tex] and [tex]\( (5, -3)\)[/tex], we need to follow these steps:
1. Find the slope (m) of the line: The slope [tex]\(m\)[/tex] is calculated using the formula:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Here, the points are [tex]\((x_1, y_1) = (-4, 7)\)[/tex] and [tex]\((x_2, y_2) = (5, -3)\)[/tex].
Plugging in the values, we get:
[tex]\[
m = \frac{-3 - 7}{5 + 4} = \frac{-10}{9}
\][/tex]
2. Determine the point-slope form: The point-slope form of the equation of a line is given by:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
3. Substitute one of the points and the slope into the point-slope form:
We use the point [tex]\((-4, 7)\)[/tex] and the slope [tex]\(m = -\frac{10}{9}\)[/tex].
Substituting these values into the point-slope form, we have:
[tex]\[
y - 7 = -\frac{10}{9}(x - (-4))
\][/tex]
4. Simplify the equation:
Since [tex]\(x - (-4)\)[/tex] simplifies to [tex]\(x + 4\)[/tex], the equation becomes:
[tex]\[
y - 7 = -\frac{10}{9}(x + 4)
\][/tex]
So, the point-slope form of the equation for the line passing through the points [tex]\((-4, 7)\)[/tex] and [tex]\((5, -3)\)[/tex] is:
[tex]\[
y - 7 = -\frac{10}{9}(x + 4)
\][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{D} \][/tex]
