Background for slopes:

#color(white)("XXXX")#The slope of a line is defined as

#color(white)("XXXX")##color(white)("XXXX")##(Delta y)/(Delta x)#

#color(white)("XXXX")#That is, given two points #(x_1,y_1)# and #(x_2,y_2)# on the line

#color(white)("XXXX")#the slope is

#color(white)("XXXX")##color(white)("XXXX")##m=(y_2-y_1)/(x_2-x_1)#

#color(white)("XXXX")#For a straight line the slope is the same for all pairs of points on the line

#color(white)("XXXX")#Therefore, given two fixed points (as above) and a variable point #(x,y)# on the line

#color(white)("XXXX")##color(white)("XXXX")##(y-y_1)/(x-x_1) = (y_2-y_1)/(x_2-x_1)#

#color(white)("XXXX")#This can be rewritten:

#color(white)("XXXX")##color(white)("XXXX")##y=m(x-x_1)+y_1#

#color(white)("XXXX")#If a line has a slope of #hatm# then all lines perpendicular to it have a slope of #1/(hatm)#

Slope of #6x-7y=6#

#color(white)("XXXX")#This equation can be rewritten as

#color(white)("XXXX")##color(white)("XXXX")##y = (6/7)x+(6/7)#

#color(white)("XXXX")#and therefore has a slope of #(6/7)#

#color(white)("XXXX")#Any line perpendicular to it has a slope of #(-7/6)#

Equation of a line through #(2,3)# perpendicular to #6x-7y=6#

#color(white)("XXXX")#Using the previous discussion:

#color(white)("XXXX")##color(white)("XXXX")##y = (-7/6)(x-2)+3#

#color(white)("XXXX")#or, simplified and re-written in function notation

#color(white)("XXXX")##color(white)("XXXX")##f(x) = -7/6x+2/3#