To solve the problem of finding the equation of a line perpendicular to line [tex]\(LM\)[/tex] with the given equation [tex]\(y = 5x + 4\)[/tex] that passes through the point [tex]\((-3, 2)\)[/tex], let's proceed with the following steps:
1. Determine the slope of line [tex]\(LM\)[/tex]:
The given equation of line [tex]\(LM\)[/tex] is [tex]\(y = 5x + 4\)[/tex]. The slope-intercept form of a line is [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] represents the slope. From the equation, we can see that the slope [tex]\(m\)[/tex] of line [tex]\(LM\)[/tex] is 5.
2. Find the slope of the perpendicular line:
Two lines are perpendicular if the product of their slopes is [tex]\(-1\)[/tex]. Therefore, if one line has a slope [tex]\(m\)[/tex], the perpendicular line will have a slope of [tex]\(-\frac{1}{m}\)[/tex].
Thus, the slope [tex]\(m_\perp\)[/tex] of the line perpendicular to [tex]\(LM\)[/tex] is:
[tex]\[
m_\perp = -\frac{1}{5}
\][/tex]
3. Use the point-slope form to find the equation of the perpendicular line:
The point-slope form of the equation of a line 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 are given the point [tex]\((-3, 2)\)[/tex] and we have found [tex]\(m_\perp = -\frac{1}{5}\)[/tex]. Plugging in these values, we get:
[tex]\[
y - 2 = -\frac{1}{5}(x + 3)
\][/tex]
4. Simplify the equation to slope-intercept form:
First, distribute the slope on the right-hand side:
[tex]\[
y - 2 = -\frac{1}{5}x - \frac{3}{5}
\][/tex]
Next, add 2 to both sides to solve for [tex]\(y\)[/tex]:
[tex]\[
y = -\frac{1}{5}x - \frac{3}{5} + 2
\][/tex]
Convert 2 to a fraction with the same denominator:
[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]
Thus, the equation of the line perpendicular to [tex]\(LM\)[/tex] and passing through the point [tex]\((-3, 2)\)[/tex] is:
[tex]\[
\boxed{y = -\frac{1}{5}x + \frac{7}{5}}
\][/tex]
