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)(3))/(color(red)(2) - color(blue)(-1)) = (color(red)(-3) - color(blue)(3))/(color(red)(2) + color(blue)(1)) = -6/3 = -2#
We can now use the point-slope formula to write the equation for a 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 gives:
#(y - color(red)(3)) = color(blue)(-2)(x - color(red)(-1))#
#(y - color(red)(3)) = color(blue)(-2)(x + color(red)(1))#
We can also substitute the slope we calculated and the second point giving:
#(y - color(red)(-3)) = color(blue)(-2)(x - color(red)(2))#
#(y + color(red)(3)) = color(blue)(-2)(x - color(red)(2))#
We can also solve this equation for #y# 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)(3) = (color(blue)(-2) xx x) - (color(blue)(-2) xx color(red)(2))#
#y + color(red)(3) = -2x + 4#
#y + color(red)(3) - 3 = -2x + 4 - 3#
#y + 0 = -2x + 1#
#y = color(red)(-2)x + color(blue)(1)#
Three equations which solve this problem are:
#(y - color(red)(3)) = color(blue)(-2)(x + color(red)(1))#
Or
#(y + color(red)(3)) = color(blue)(-2)(x - color(red)(2))#
Or
#y = color(red)(-2)x + color(blue)(1)#
