Given the points #(-3,5)# and #(-6,8)#
Step 1: Determine the slope
The slope of a line between two points is the change in the y values divided by change in the x values. That is, given two point #(x_1,y_1)# and #(x_2,y_2)#, the slope is:
#color(white)("XXXX")##m = (Delta y)/(Delta x) = (y_2 - y_1)/(x_2 - x_1)#
For the given example this becomes:
#color(white)("XXXX")##m = (8 - 5)/((-6) - (-3))#
#color(white)("XXXX")##color(white)("XXXX")##= 3/(-3) = -1#
Step 2: Combine the slope with a point on the line for the equation
The point-slope form for a linear equation with
#color(white)("XXXX")#slope #m#
#color(white)("XXXX")#through a point #(hatx, haty)#
is
#color(white)("XXXX")##y-haty = m(x-hatx)#
For the given example we could choose either of the given point to use as #(hatx, haty)#. For demonstration purposes I have used #(hatx, haty) = (-3,5)#
We have already determined that #m = -1#
So a point-slope form of the line would be:
#color(white)("XXXX")##y-5 = (-1)(x-(-3))#
#color(white)("XXXX")#[You might simplify this as:
#color(white)("XXXX")##color(white)("XXXX")##y-5 = -x-3#
#color(white)("XXXX")#or
#color(white)("XXXX")##color(white)("XXXX")##y = 2 -x#
#color(white)("XXXX")#but, while both have a simpler appearance,
#color(white)("XXXX")#they are not true "point-slope forms".]
