For the line segment from #(-0.5,0.9)# to #(-3.3,-0.5)#
the change in #y# is #delta y = ((-0.5)-(0.9))=-1.4#
and
the change in #x# is #delta x = ((-3.3)-(-0.5))=-2.8#
Slope is defined as #color(green)m=(delta y)/(delta x)#
So, in this case,
#color(white)("XXX")#the slope is #color(green)(m)=(-1.4)/(-2.8)=color(green)(1/2)#
For a line with a slope of #color(green)(m)# through a point #(color(red)(x_0),color(blue)(y_0))#
the point-slope form of the equation is
#color(white)("XXX")y-color(blue)(y_0)=color(green)(m)(x-color(red)(x_0))#
We could use either of the given points as #(color(red)(x_0),color(blue)(y_0))#
As an example if we choose #(color(red)(x_0),color(blue)(y_0)) = (color(red)(-0.5),color(blue)(0.9))#
then the equation of our line would be
#color(white)("XXX")y-color(blue)(0.9)=color(green)(1/2)(x-color(red)((-0.5)))#
or
#color(white)("XXX")y-0.9=1/2(x+0.5)#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Alternately, if we had chosen #(color(red)(x_0),color(blue)(y_0)) = (color(red)(-3.3),color(blue)(-0.5))#
the resulting equation would be
#color(white)("XXX")y+0.5=1/2(x+3.3)#
With a bit of manipulation these equations can be shown to be identical.
