First, we need to solve the equation in the problem for #y# to put it in slope-intercept form so we can determine its slope:
#2y - 6x = 4#
#2y - 6x + color(red)(6x) = color(red)(6x) + 4#
#2y - 0 = 6x + 4#
#2y = 6x + 4#
#(2y)/color(red)(2) = (6x + 4)/color(red)(2)#
#(color(red)(cancel(color(black)(2)))y)/cancel(color(red)(2)) = ((6x)/color(red)(2)) + (4/color(red)(2))#
#y = 3x + 2#
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.
Therefore the slope of this equation is #color(red)(m = 3)#
A perpendicular line will have a slope (let's call this slope #m_p#) that is the negative inverse of this line. Or, #m_p = -1/m#
Substituting gives:
#m_p = -1/3#
