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)(-3) - color(blue)(15))/(color(red)(11) - color(blue)(21)) = (-18)/-10 = 9/5#
We can now 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 values from the first point in the problem gives:
Solution 1: #(y - color(red)(15)) = color(blue)(9/5)(x - color(red)(21))#
We can also substitute the slope we calculated and the values from the second point in the problem giving:
#(y - color(red)(-3)) = color(blue)(9/5)(x - color(red)(11))#
Solution 2: #(y + color(red)(3)) = color(blue)(9/5)(x - color(red)(11))#
We can also solve the first equation for #y# to put 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)(15) = (color(blue)(9/5) * x) - (color(blue)(9/5) * color(red)(21))#
#y - color(red)(15) = 9/5x - 189/5#
#y - color(red)(15) + 15 = 9/5x - 189/5 + 15#
#y - 0 = 9/5x - 189/5 + (5/5 xx 15)#
#y = 9/5x - 189/5 + 75/5#
Solution 3: #y = color(red)(9/5)x - color(blue)(114/5)#
