First, we need to determine the slope of the line which passes through the two points in the problem. 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)(3/4) - color(blue)(1/2))/(color(red)(1/4) - color(blue)(-1/2)) = (color(red)(3/4) - color(blue)(1/2))/(color(red)(1/4) + color(blue)(1/2)) = (color(red)(3/4) - (2/2 xx color(blue)(1/2)))/(color(red)(1/4) + (2/2 xx color(blue)(1/2))) =#
#(color(red)(3/4) - color(blue)(2/4))/(color(red)(1/4) + color(blue)(2/4)) = (1/4)/(3/4) = (1 xx 4)/(4 xx 3) = 4/12 = 1/3#
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 values from the first point in the problem gives:
#(y - color(red)(1/2)) = color(blue)(1/3)(x - color(red)(-1/2))#
#(y - color(red)(1/2)) = color(blue)(1/3)(x + color(red)(1/2))#
We can also substitute the slope we calculated and the values from the second point in the problem giving:
#(y - color(red)(3/4)) = color(blue)(1/3)(x - color(red)(1/4))#
