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)(-4) - color(blue)(2))/(color(red)(3) - color(blue)(-1)) = (color(red)(-4) - color(blue)(2))/(color(red)(3) + color(blue)(1)) = -6/4 = -3/2#
Next we can use the point-slope formula to write 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 values from the second point in the problem gives:
#(y - color(red)(-4)) = color(blue)(-3/2)(x - color(red)(3))#
#(y + color(red)(4)) = color(blue)(-3/2)(x - color(red)(3))#
We can also substitute the slope we calculated and the values from the first point in the problem giving:
#(y - color(red)(2)) = color(blue)(-3/2)(x - color(red)(-1))#
#(y - color(red)(2)) = color(blue)(-3/2)(x + color(red)(1))#
We can solve this equation for #y# to write an 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)(2) = (color(blue)(-3/2) xx x) + (color(blue)(-3/2) xx color(red)(1))#
#y - color(red)(2) = -3/2x + (-3/2)#
#y - color(red)(2) = -3/2x - 3/2#
#y - color(red)(2) + 2 = -3/2x - 3/2 + 2#
#y - 0 = -3/2x - 3/2 + (2/2 xx 2)#
#y = -3/2x - 3/2 + 4/2#
#y = color(red)(-3/2)x + color(blue)(1/2)#
