First, we need to determine the slope of 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 values from the points in the problem gives:
#m = (color(red)(9) - color(blue)(-8))/(color(red)(-3) - color(blue)(-1)) = (color(red)(9) + color(blue)(8))/(color(red)(-3) + color(blue)(1)) = 17/-2 = -17/2#
We can now use the point slope formula to write an equation for the line. 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.
Substituting the slope we calculated and the values first point in the problem gives:
#(y - color(blue)(-8)) = color(red)(-17/2)(x - color(blue)(-1))#
#(y + color(blue)(8)) = color(red)(-17/2)(x + color(blue)(1))#
We can also substitute the slope and the values from the second point in the problem giving:
#(y - color(blue)(9)) = color(red)(-17/2)(x - color(blue)(-3))#
#(y - color(blue)(9)) = color(red)(-17/2)(x + color(blue)(3))#
We can transform this equation into the 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(blue)(9) = (color(red)(-17/2) xx x) + (color(red)(-17/2) xx color(blue)(3))#
#y - color(blue)(9) = -17/2x + (-51/2)#
#y - color(blue)(9) = -17/2x - 51/2#
#y - color(blue)(9) + 9 = -17/2x - 51/2 + 9#
#y - 0 = -17/2x - 51/2 + 18/2#
#y = color(red)(-17/2)x - color(blue)(33/2)#
