Because both sides of the equation are pure fractions we can flip the fractions without impacting the equality:
#-(f + 4)/5 = -(f + 8)/1#
#-(f + 4)/5 = -(f + 8)#
Next, multiply each side of the equation by #color(red)(-5)# to eliminate the fraction while keeping the equation balanced:
#color(red)(-5) xx (f + 4)/-5 = color(red)(-5) xx -(f + 8)#
#cancel(color(red)(-5)) xx (f + 4)/color(red)(cancel(color(black)(-5))) = color(red)(5) xx (f + 8)#
#f + 4 = (color(red)(5) xx f) + (color(red)(5) xx 8)#
#f + 4 = 5f + 40#
Then, subtract #color(red)(f)# and #color(blue)(40)# from each side of the equation to isolate the #f# term while keeping the equation balanced:
#f - color(red)(f) + 4 - color(blue)(40) = 5f - color(red)(f) + 40 - color(blue)(40)#
#0 - 36 = 5f - color(red)(1f) + 0#
#-36 = (5 - color(red)(1))f#
#-36 = 4f#
Now, divide each side of the equation by #color(red)(4)# to solve for #f# while keeping the equation balanced:
#(-36)/color(red)(4) = 4f/color(red)(4)#
#-9 = color(red)(cancel(color(black)(4)))f/cancel(color(red)(4))#
#-9 = f#
#f = -9#
Validating the solution gives:
#-5/(f + 4) = -1/(f + 8)# becomes:
#-5/(-9 + 4) = -1/(-9 + 8)#
#-5/(-5) = -1/(-1)#
#1 = 1#
