To find the equation of the line that is perpendicular to the given line and passes through a specific point, we can follow a series of steps. Here is the detailed, step-by-step solution.
1. Identify the slope of the given line:
The given equation of the line is in point-slope form: [tex]\(y + 3 = -4(x + 4)\)[/tex].
In point-slope form, the equation of a line is [tex]\(y - y_1 = m(x - x_1)\)[/tex], where [tex]\(m\)[/tex] is the slope.
From the given equation, we can identify the slope [tex]\(m\)[/tex] as [tex]\(-4\)[/tex].
2. Find the slope of the perpendicular line:
The slope of a line that is perpendicular to another line is the negative reciprocal of the slope of the original line.
Since the slope of the given line is [tex]\(-4\)[/tex], the slope of the perpendicular line will be the negative reciprocal of [tex]\(-4\)[/tex].
The negative reciprocal of [tex]\(-4\)[/tex] is [tex]\(\frac{1}{4}\)[/tex].
3. Use the point-slope form to write the equation of the perpendicular line:
We are given that the perpendicular line passes through the point [tex]\((-4, -3)\)[/tex].
The point-slope form of a line is given by [tex]\(y - y_1 = m(x - x_1)\)[/tex], where:
- [tex]\(m\)[/tex] is the slope of the line
- [tex]\((x_1, y_1)\)[/tex] is a point on the line
Here, [tex]\(m = \(\frac{1}{4}\)[/tex]\) and the point [tex]\((x_1, y_1) = (-4, -3)\)[/tex].
4. Substitute the point and the slope into the point-slope form equation:
[tex]\[ y - (-3) = \frac{1}{4}(x - (-4)) \][/tex]
Simplify the equation:
[tex]\[ y + 3 = \frac{1}{4}(x + 4) \][/tex]
So, the equation, in point-slope form, of the line that is perpendicular to the given line and passes through the point [tex]\((-4, -3)\)[/tex] is:
[tex]\[ y + 3 = \frac{1}{4}(x + 4) \][/tex]
This matches the last option given:
[tex]\[ y + 3 = \frac{1}{4}(x + 4) \][/tex]
