First, we will write and equation in point-slope form and then convert to slope-intercept form.
To use the point-slope form we must first determine the slope.
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)(4))#
#m = (color(red)(3) + color(blue)(3))/(color(red)(2) - color(blue)(4))#
#m = 6/-2 = -3#
We can now use this calculated slope and either point to write the equation in point-slope form.
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.
Again, substituting gives:
#(y - color(red)(-3)) = color(blue)(-3)(x - color(red)(4))#
#(y + color(red)(3)) = color(blue)(-3)(x - color(red)(4))#
We can now convert this to 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.
We can solve our equation for #y#:
#(y + color(red)(3)) = color(blue)(-3)(x - color(red)(4))#
#y + color(red)(3) = (color(blue)(-3) xx x) - (color(blue)(-3) xx color(red)(4))#
#y + 3 = -3x - (-12)#
#y + 3 = -3x + 12#
#y + 3 - color(red)(3) = -3x + 12 - color(red)(3)#
#y + 0 = -3x + 9#
#y = -3x + 9#
