The point-slope formula can be used to find this equation. However, we must first find the slope which can be found using two points on a 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 problem gives:
#m = (color(red)(2) - color(blue)(-6))/(color(red)(4) - color(blue)(5))#
#m = (color(red)(2) + color(blue)(6))/(color(red)(4) - color(blue)(5))#
#m = 8/-1 = -8#
The slope and either of the points can now be used with 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 calculate slope and the second point gives:
#(y - color(red)(2)) = color(blue)(-8)(x - color(red)(4))#
Or, we can convert to the more familiar slope-intercept form by solving for #y#:
#y - color(red)(2) = (color(blue)(-8) xx x) - (color(blue)(-8) xx color(red)(4))#
#y - 2 = -8x + 32#
#y - 2 + color(red)(2) = -8x + 32 + color(red)(2)#
#y - 0 = -8x + 34#
#y = -8x + 34#
Or, we can use the point-slope formula and the first point to give:
#(y - color(red)(-6)) = color(blue)(-8)(x - color(red)(5))#
#(y + color(red)(6)) = color(blue)(-8)(x - color(red)(5))#
