The equation in the problem is in slope-intercept form. The slope-intercept form of a linear equation is: #y = color(red)(m)x + color(blue)(b)#
Where #color(red)(m)# is the slope and #color(blue)(b)# is the y-intercept value.
#y = color(red)(-1/3)x + color(blue)(9)# so we know the slope of this line is #color(red)(m = -1/3)#
The slope of a perpendicular line (let's call it #m_p#) is the negative inverse of the slope of this line or #m_p = -1/m#
Therefore the slope of a perpendicular line is #m_p = 3#
We can now use the point-slope formula to find an equation for the line. The point-slope formula states: #(y - color(red)(y_1)) = color(blue)(m)(x - color(red)(x_1))#
Where #color(blue)(m)# is the slope and #color(red)(((x_1, y_1)))# is a point the line passes through.
Substituting the point from the problem and the slope we determined gives:
#(y - color(red)(-2)) = color(blue)(3)(x - color(red)(-2))#
#(y + color(red)(2)) = color(blue)(3)(x + color(red)(2))#
We can now solve this for #y# to put it into slope-intercept form:
#y + color(red)(2) = (color(blue)(3) xx x) + (color(blue)(3) xx color(red)(2))#
#y + color(red)(2) = 3x + 6#
#y + color(red)(2) - 2 = 3x + 6 - 2#
#y + 0 = 3x + 4#
#y = color(red)(3)x + color(blue)(4)#
