To find the equation of the line that is perpendicular to the given line and passes through the specific point, follow these steps:
### Step 1: Identify the slope of the given line
The given equation of the line is in point-slope form:
[tex]\[ y - 3 = -4(x + 2) \][/tex]
From this, you can see that the slope ([tex]\(m\)[/tex]) of the given line is [tex]\(-4\)[/tex].
### Step 2: Determine the slope of the perpendicular line
For two lines to be perpendicular, the product of their slopes must be [tex]\(-1\)[/tex]. If the slope of the given line is [tex]\(-4\)[/tex], then the slope of the line that is perpendicular to it, [tex]\(m_{\perpendicular}\)[/tex], will be the negative reciprocal of [tex]\(-4\)[/tex].
[tex]\[
m_{\perpendicular} = \frac{1}{-(-4)} = \frac{1}{4}
\][/tex]
### Step 3: Use the point-slope form of a line equation
Now that we have the slope of the perpendicular line ([tex]\(\frac{1}{4}\)[/tex]) and a point it passes through ([tex]\(-5, 7\)[/tex]), we can use the point-slope form of the equation of a line:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Here, [tex]\((x_1, y_1)\)[/tex] is [tex]\((-5, 7)\)[/tex] and [tex]\(m\)[/tex] is [tex]\(\frac{1}{4}\)[/tex].
### Step 4: Substitute the given point and slope into the point-slope formula
Substitute [tex]\((x_1, y_1) = (-5, 7)\)[/tex] and [tex]\(m = \frac{1}{4}\)[/tex] into the formula:
[tex]\[
y - 7 = \frac{1}{4}(x + 5)
\][/tex]
This is the equation of the line that is perpendicular to the given line and passes through the point [tex]\((-5, 7)\)[/tex].
### Conclusion
The correct equation that represents the perpendicular line is:
[tex]\[
y - 7 = \frac{1}{4}(x + 5)
\][/tex]
Therefore, the correct option is:
A. [tex]\(y-7=\frac{1}{4}(x+5)\)[/tex]
