We can first write an equation in point-slope form. To do this we must first determine the slope. The slope can be found by using the formula: #m = (color(red)(y_2) - color(blue)(y_1))/(color(red)(x_2) - color(blue)(x_1))#
Where #m# is the slope and (#color(blue)(x_1, y_1)#) and (#color(red)(x_2, y_2)#) are the two points on the line.
Substituting the values from the points in the problem gives:
#m = (color(red)(3) - color(blue)(-5))/(color(red)(-1) - color(blue)(-2))#
#m = (color(red)(3) + color(blue)(5))/(color(red)(-1) + color(blue)(2))#
#m = 8/1 = 8#
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 slope we calculated and the first point gives:
#(y - color(red)(-5)) = color(blue)(8)(x - color(red)(-2))#
#(y + color(red)(5)) = color(blue)(8)(x + color(red)(2))#
The standard form of a linear equation is: #color(red)(A)x + color(blue)(B)y = color(green)(C)#
where, if at all possible, #color(red)(A)#, #color(blue)(B)#, and #color(green)(C)#are integers, and A is non-negative, and, A, B, and C have no common factors other than 1
We can now transform the point-slope form to the standard form:
#y + color(red)(5) = (color(blue)(8) xx x) + (color(blue)(8) xx color(red)(2))#
#y + 5 = 8x + 16#
#color(red)(-8x) + y + 5 - color(blue)(5) = color(red)(-8x) + 8x + 16 - color(blue)(5)#
#-8x + y + 0 = 0 + 11#
#-8x + y = 11#
#color(red)(-1)(-8x + y) = color(red)(-1) xx 11#
#8x - y = -11#
