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)(-6) - color(blue)(-1))/(color(red)(-2) - color(blue)(1)) = (color(red)(-6) + color(blue)(1))/(color(red)(-2) - color(blue)(1)) = (-5)/(-3) = 5/3#
Now, we can use the point-slope formula to write an equation for the line passing through the two points. 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 gives:
#(y - color(red)(-1)) = color(blue)(5/3)(x - color(red)(1))#
#(y + color(red)(1)) = color(blue)(5/3)(x - color(red)(1))#
We can also substitute the slope we calculated and the second point giving:
#(y - color(red)(-6)) = color(blue)(5/3)(x - color(red)(-2))#
#(y + color(red)(6)) = color(blue)(5/3)(x + color(red)(2))#
We can also solve this 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)(6) = (color(blue)(5/3) xx x) + (color(blue)(5/3) xx color(red)(2))#
#y + color(red)(6) = 5/3x + 10/3#
#y + color(red)(6) - 6 = 5/3x + 10/3 - 6#
#y + 0 = 5/3x + 10/3 - (3/3 xx 6)#
#y = 5/3x + 10/3 - 18/3#
#y = color(red)(5/3)x - color(blue)(8/3)#
