First, we must 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)(-7) - color(blue)(9))/(color(red)(3) - color(blue)(-7)) = (color(red)(-7) - color(blue)(9))/(color(red)(3) + color(blue)(7)) = -16/10 = -8/5#
Now, we can use the point-slope formula to write and 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 the first point from the problem gives:
#(y - color(red)(9)) = color(blue)(-8/5)(x - color(red)(-7))#
Solution 1: #(y - color(red)(9)) = color(blue)(-8/5)(x + color(red)(7))#
We can also substitute the slope we calculated and the second point from the problem giving:
#(y - color(red)(-7)) = color(blue)(-8/5)(x - color(red)(3))#
Solution 2: #(y + color(red)(7)) = color(blue)(-8/5)(x - color(red)(3))#
Or, we can solve the first or second equation for #y# and write the equation in slope intercept form. The slope-intercept form of a linear equation is: #y = color(red)(m)x + color(blue)(b)#
Where #color(red)(m)# is the slope and #color(blue)(b)# is the y-intercept value.
#y + color(red)(7) = (color(blue)(-8/5) xx x) - (color(blue)(-8/5) xx color(red)(3))#
#y + color(red)(7) = -8/5x - (-24/5)#
#y + color(red)(7) = -8/5x + 24/5#
#y + color(red)(7) - 7 = -8/5x + 24/5 - 7#
#y + 0 = -8/5x + 24/5 - (5/5 xx 7)#
#y + 0 = -8/5x + 24/5 - 35/5#
Solution 3: #y = color(red)(-8/5)x - color(blue)(11/5)#
