Because we are given two points we can calculate the slope and then we can use the slope and either point and use the point-slope formula to find the equation for the 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.
Substituting the two points from the problem gives:
#m = (color(red)(-9) - color(blue)(-3))/(color(red)(-4) - color(blue)(6))#
#m = (color(red)(-9) + color(blue)(3))/(color(red)(-4) - color(blue)(6))#
#m = (-6)/(-10)#
#m = 6/10 = 3/5#
Now we have the slope and can use either point and the point-slope formula to find the equation for the line:
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 one of the points gives:
#(y - color(red)(-9)) = color(blue)(3/5)(x - color(red)(-4))#
#(y + color(red)(9)) = color(blue)(3/5)(x + color(red)(4))#
We can convert to the more familiar slope-intercept form by solving for #y#:
#y + color(red)(9) = color(blue)(3/5)x + (color(blue)(3/5) xx color(red)(4))#
#y + color(red)(9) = color(blue)(3/5)x + 12/5#
#y + color(red)(9) - 9 = color(blue)(3/5)x + 12/5 - 9#
#y + 0 = color(blue)(3/5)x + 12/5 - (9 xx 5/5)#
#y = color(blue)(3/5)x + 12/5 - 45/5#
#y = color(blue)(3/5)x - 33/5#
