Certainly! Let's solve this step-by-step.
1. Identify the given line's slope:
The equation provided is [tex]\(y - 3 = -4(x + 2)\)[/tex]. This is in the point-slope form [tex]\((y - y_1 = m(x - x_1))\)[/tex], where [tex]\(m\)[/tex] is the slope. From this equation, we identify the slope ([tex]\(m\)[/tex]) of the given line as [tex]\(-4\)[/tex].
2. Find the slope of the perpendicular line:
The slope of a line perpendicular to another line is the negative reciprocal of the slope of the given line. Since the slope of the given line is [tex]\(-4\)[/tex], the negative reciprocal of [tex]\(-4\)[/tex] is [tex]\(\frac{1}{4}\)[/tex].
3. Write the equation of the perpendicular line:
We need to find the equation of the line that is perpendicular to the given line and passes through the point [tex]\((-5, 7)\)[/tex].
Using the point-slope form again, [tex]\((y - y_1 = m(x - x_1))\)[/tex], where [tex]\(m\)[/tex] is the slope of the perpendicular line, and [tex]\((x_1, y_1)\)[/tex] is the point [tex]\((-5, 7)\)[/tex]:
[tex]\[
y - 7 = \frac{1}{4}(x + 5)
\][/tex]
4. Match the equation to the options provided:
Let's compare the derived equation [tex]\(y - 7 = \frac{1}{4}(x + 5)\)[/tex] with the given options:
A. [tex]\(y - 7 = -4(x + 5)\)[/tex]
B. [tex]\(y + 7 = -\frac{1}{4}(x - 5)\)[/tex]
C. [tex]\(y + 7 = 4(x - 5)\)[/tex]
D. [tex]\(y - 7 = \frac{1}{4}(x + 5)\)[/tex]
The correct option is:
[tex]\[
\text{D. } y - 7 = \frac{1}{4}(x + 5)
\][/tex]
To summarize, the equation of the line that is perpendicular to [tex]\(y - 3 = -4(x + 2)\)[/tex] and passes through the point [tex]\((-5, 7)\)[/tex] is [tex]\(y - 7 = \frac{1}{4}(x + 5)\)[/tex].
The correct answer is [tex]\( \text{D. } y - 7 = \frac{1}{4}(x + 5) \)[/tex].
