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)(1) - color(blue)(-5))/(color(red)(-2) - color(blue)(3)) = (color(red)(1) + color(blue)(5))/(color(red)(-2) - color(blue)(3)) = 6/-5 = -6/5#
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)(-5)) = color(blue)(-6/5)(x - color(red)(3))#
#(y + color(red)(5)) = color(blue)(-6/5)(x - color(red)(3))#
We can also substitute the slope we calculated and the second point from the problem giving:
#(y - color(red)(1)) = color(blue)(-6/5)(x - color(red)(-2))#
#(y - color(red)(1)) = color(blue)(-6/5)(x + color(red)(2))#
We can solve this equation to put the formula 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)(1) = (color(blue)(-6/5) xx x) + (color(blue)(-6/5) xx color(red)(2))#
#y - color(red)(1) = -6/5x + (-12/5)#
#y - color(red)(1) = -6/5x - 12/5#
#y - color(red)(1) + 1 = -6/5x - 12/5 + 1#
#y - 0 = -6/5x - 12/5 + (1 xx 5/5)#
#y = -6/5x - 12/5 + 5/5#
#y = color(red)(-6/5)x - color(blue)(7/5)#
