Answer:
[tex]y = -\dfrac{5}{3} x - 9[/tex]
Step-by-step explanation:
Start by finding the slope of the perpendicular line. The slope is the negative reciprocal of the original line. From there, use the point-slope form of a line and plug in the point and the perpendicular slope and evaluate to get the whole expression in slope-intercept form.
[tex]\begin{document}\fbox{% \begin{minipage}{0.9\textwidth} Point-slope formula: \[ y - y_1 = m(x - x_1) \] Where: \begin{itemize} \item \( y \): dependent variable \item \( y_1 \): value at point \item \( m \): slope \item \( x \): independent variable \item \( x_1 \): value at point \end{itemize} \end{minipage}}\end{document}[/tex]
Solving:
[tex]\subsection*{Slope of the Perpendicular Line:}\\Take the negative reciprocal of the original line:\\\[m_2 = -\frac{1}{m_1} = -\frac{1}{\frac{3}{5}} = \boxed{-\frac{5}{3}}\][/tex]
[tex]\subsection*{Plug into Point-Slope Equation:}The point-slope form:\[y - y_1 = m(x - x_1)\]\\Plug in the values:\[y - (-4) = -\frac{5}{3}(x - (-3))\]\[\boxed{y + 4 = -\frac{5}{3}(x + 3)}\][/tex]
[tex]\subsection*{Convert to Slope-Intercept Form:}\[y + 4 = -\frac{5}{3}x - \frac{5}{3} \times 3\]\[y + 4 = -\frac{5}{3}x - 5\]\[y = -\frac{5}{3}x - 5 - 4\]\[\boxed{y = -\frac{5}{3}x - 9}\][/tex]
This is the equation of the perpendicular line.