First, we must 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)(-5) - color(blue)(1))/(color(red)(-2) - color(blue)(-3)) = (color(red)(-5) - color(blue)(1))/(color(red)(-2) + color(blue)(3)) = -6/1 = -6#
We can now use the point slope formula to find an 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)(1)) = color(blue)(-6)(x - color(red)(-3))#
Solution 1) #(y - color(red)(1)) = color(blue)(-6)(x + color(red)(3))#
We can also substitute the slope we calculated and the second point from the problem giving:
#(y - color(red)(-5)) = color(blue)(-6)(x - color(red)(-2))#
Solution 2) #(y + color(red)(5)) = color(blue)(-6)(x + color(red)(2))#
We can also solve this 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)(5) = (color(blue)(-6) xx x) + (color(blue)(-6) xx color(red)(2))#
#y + color(red)(5) = -6x - 12#
#y + color(red)(5) - 5 = -6x - 12 - 5#
#y + 0 = -6x - 17#
Solution 3) #y = color(red)(-6)x - color(blue)(17)#
