To find the equation of a line that is perpendicular to the given line \( y + 3 = -4(x + 4) \) and passes through the point \((-4, -3)\), follow these steps:
1. Determine the slope of the given line:
The given line's equation is \( y + 3 = -4(x + 4) \). This equation is in point-slope form \( y - y_1 = m(x - x_1) \), where \( m \) is the slope. Here, it is clear that the slope \( m \) of the given line is \(-4\).
2. Find the slope of the perpendicular line:
The slope of a line perpendicular to another is the negative reciprocal of the given line's slope. So, the negative reciprocal of \(-4\) is \(\frac{1}{4}\). Hence, the slope of the perpendicular line is \(\frac{1}{4}\).
3. Use the point-slope form to write the equation of the new line:
Point-slope form of a line's equation is \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is a point on the line and \( m \) is the slope.
Plugging in the given point \((-4, -3)\) and the slope \(\frac{1}{4}\) into the point-slope form:
[tex]\[
y - (-3) = \frac{1}{4}(x - (-4))
\][/tex]
Simplifying, we get:
[tex]\[
y + 3 = \frac{1}{4}(x + 4)
\][/tex]
Therefore, the equation of the line that is perpendicular to the given line and passes through the point \((-4, -3)\) is:
[tex]\[
y + 3 = \frac{1}{4}(x + 4)
\][/tex]
Thus, the correct choice is:
[tex]\[ \boxed{y+3=\frac{1}{4}(x+4)} \][/tex]
