The slope-intercept form of a line is:
y = mx + b
Where m is the slope and b is the y-intercept.
Two lines having slopes m1 and m2 are perpendicular if
m1 * m2 = -1
We are given the equation of one line:
[tex]y=\frac{1}{3}x-5[/tex]It's required to find the equation of another line that is perpendicular to it. We have m1 = 1/3, let's find the other slope:
[tex]\begin{gathered} m_1\cdot m_2=-1 \\ \frac{1}{3}\cdot m_2=-1 \\ Thus\colon \\ m_2=-3 \end{gathered}[/tex]The slope of the required line is -3. Now we use the point through which the line passes (-2, 2).
We use the point-slope form of the line:
y - k = m( x - h)
Where (h,k) = (-2, 2) is the point. Substituting:
[tex]y-2=-3(x+2)[/tex]Operating:
[tex]\begin{gathered} y-2=-3x-6 \\ \text{Adding 2:} \\ \boxed{y=-3x-4^{}} \end{gathered}[/tex]