To find the equation of a line perpendicular to the given line [tex]\( y = \frac{1}{3}x + 4 \)[/tex] that passes through the point [tex]\((-3, 9)\)[/tex], we can follow these steps:
1. Determine the slope of the given line:
The given line is [tex]\( y = \frac{1}{3}x + 4 \)[/tex]. The slope ([tex]\(m_1\)[/tex]) of this line is [tex]\( \frac{1}{3} \)[/tex].
2. Find the slope of the perpendicular line:
Perpendicular lines have slopes that are negative reciprocals of each other. Therefore, if the slope of the given line is [tex]\( \frac{1}{3} \)[/tex], then the slope ([tex]\(m_2\)[/tex]) of the line perpendicular to it is:
[tex]\[
m_2 = -\frac{1}{ \left( \frac{1}{3} \right) } = -3
\][/tex]
3. Use the point-slope form of the equation of a line:
The point-slope form of a line's equation is given by:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
where [tex]\(m\)[/tex] is the slope and [tex]\((x_1, y_1)\)[/tex] is a point through which the line passes. Using the point [tex]\((-3, 9)\)[/tex] and the slope [tex]\(-3\)[/tex]:
[tex]\[
y - 9 = -3(x - (-3))
\][/tex]
Simplifying inside the parentheses:
[tex]\[
y - 9 = -3(x + 3)
\][/tex]
4. Expand and simplify to the slope-intercept form:
Distribute the slope [tex]\(-3\)[/tex]:
[tex]\[
y - 9 = -3x - 9
\][/tex]
Add 9 to both sides to solve for [tex]\(y\)[/tex]:
[tex]\[
y = -3x
\][/tex]
Therefore, the equation of the line that is perpendicular to [tex]\( y = \frac{1}{3}x + 4 \)[/tex] and passes through the point [tex]\((-3, 9)\)[/tex] is:
[tex]\[
\boxed{y = -3x}
\][/tex]
