To find the equation of the line that is perpendicular to the given line [tex]\( y - 3 = -4(x + 2) \)[/tex] and passes through the point [tex]\((-5, 7)\)[/tex], follow these steps:
1. Find the slope of the given line:
The given line is in point-slope form [tex]\( y - y_1 = m(x - x_1) \)[/tex]. First, we'll convert it to slope-intercept form [tex]\( y = mx + b \)[/tex].
[tex]\[
y - 3 = -4(x + 2)
\][/tex]
Distribute the [tex]\(-4\)[/tex] on the right-hand side:
[tex]\[
y - 3 = -4x - 8
\][/tex]
Add 3 to both sides to isolate [tex]\(y\)[/tex]:
[tex]\[
y = -4x - 8 + 3
\][/tex]
Simplify:
[tex]\[
y = -4x - 5
\][/tex]
So, the slope [tex]\( m_1 \)[/tex] of the given line is [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 given line’s slope. The negative reciprocal of [tex]\(-4\)[/tex] is [tex]\(\frac{1}{4}\)[/tex].
3. Use the point-slope form to find the equation of the new line:
The point-slope form of a line's equation is given by [tex]\( y - y_1 = m(x - x_1) \)[/tex], where [tex]\( (x_1, y_1) \)[/tex] is a point on the line and [tex]\( m \)[/tex] is the slope. We have the point [tex]\((-5, 7)\)[/tex] and the slope [tex]\(\frac{1}{4}\)[/tex].
Substitute these values into the point-slope form:
[tex]\[
y - 7 = \frac{1}{4}(x + 5)
\][/tex]
4. Match the equation with the given options:
Simplify the equation:
[tex]\[
y - 7 = \frac{1}{4}(x + 5)
\][/tex]
This matches option [tex]\(D\)[/tex].
Therefore, 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:
D. [tex]\( y - 7 = \frac{1}{4}(x + 5) \)[/tex]
