First, let's name the three points.
#A# is #(1, -2)#; #B# is #(5, -6)#; #C# is #(0,0)#
First, let's find the slope of each line. 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.
Slope A-B:
#m_(A-B) = (color(red)(-6) - color(blue)(-2))/(color(red)(5) - color(blue)(1)) = (color(red)(-6) + color(blue)(2))/(color(red)(5) - color(blue)(1)) = -4/4 = -1#
Slope A-C:
#m_(A-C) = (color(red)(0) - color(blue)(-2))/(color(red)(0) - color(blue)(1)) = (color(red)(0) + color(blue)(2))/(color(red)(0) - color(blue)(1)) = 2/-1 = -2#
Slope B-C:
#m_(A-B) = (color(red)(0) - color(blue)(-6))/(color(red)(0) - color(blue)(5)) = (color(red)(0) + color(blue)(6))/(color(red)(0) - color(blue)(5)) = 6/-5 = -6/5#
The point-slope form of a linear equation is: #(y - color(blue)(y_1)) = color(red)(m)(x - color(blue)(x_1))#
Where #(color(blue)(x_1), color(blue)(y_1))# is a point on the line and #color(red)(m)# is the slope.
We can substitute each of the slopes we calculated and one point from each line to write an equation in point-slope form:
Line A-B:
#(y - color(blue)(-2)) = color(red)(-1)(x - color(blue)(1))#
#(y + color(blue)(2)) = color(red)(-1)(x - color(blue)(1))#
Or
#(y + color(blue)(2)) = color(red)(-)(x - color(blue)(1))#
Line A-C:
#(y - color(blue)(-2)) = color(red)(-2)(x - color(blue)(1))#
#(y + color(blue)(2)) = color(red)(-2)(x - color(blue)(1))#
Line B-C:
#(y - color(blue)(-6)) = color(red)(-6/5)(x - color(blue)(5))#
#(y + color(blue)(6)) = color(red)(-6/5)(x - color(blue)(5))#
