To find the equation of a line perpendicular to the given line [tex]\(5x - y = -4\)[/tex] and containing the point [tex]\((-3, 2)\)[/tex], we follow these steps:
1. Find the slope of line LM:
The given equation of line LM is [tex]\(5x - y = -4\)[/tex].
To find the slope, we convert the equation to the slope-intercept form [tex]\(y = mx + b\)[/tex]:
[tex]\[
5x - y = -4 \implies y = 5x + 4
\][/tex]
Here, [tex]\(m = 5\)[/tex] is the slope of line LM.
2. Determine the slope of the perpendicular line:
The slope of a line perpendicular to another line is the negative reciprocal of the original slope.
For the slope [tex]\(m = 5\)[/tex], the negative reciprocal is [tex]\(-\frac{1}{5}\)[/tex].
3. Use point-slope form to find the equation:
We know the line passes through the point [tex]\((-3, 2)\)[/tex] and has a slope of [tex]\(-\frac{1}{5}\)[/tex].
The point-slope form of a line's equation is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Plug in the point [tex]\((-3, 2)\)[/tex] and the slope [tex]\(-\frac{1}{5}\)[/tex]:
[tex]\[
y - 2 = -\frac{1}{5}(x + 3)
\][/tex]
4. Convert to slope-intercept form:
Expand and simplify to put the equation in the form [tex]\(y = mx + b\)[/tex]:
[tex]\[
y - 2 = -\frac{1}{5}(x + 3)
\][/tex]
[tex]\[
y - 2 = -\frac{1}{5}x - \frac{3}{5}
\][/tex]
[tex]\[
y = -\frac{1}{5}x - \frac{3}{5} + 2
\][/tex]
Convert 2 to a fraction with a common denominator (5):
[tex]\[
y = -\frac{1}{5}x - \frac{3}{5} + \frac{10}{5}
\][/tex]
Combine the fractions:
[tex]\[
y = -\frac{1}{5}x + \frac{7}{5}
\][/tex]
So, the equation of the line is:
[tex]\[
y = -\frac{1}{5}x + \frac{7}{5}
\][/tex]
Thus, the correct answer is [tex]\(\boxed{y = -\frac{1}{5}x + \frac{7}{5}}\)[/tex].
