How do you solve #\frac { 1} { 3} = - \frac { 7} { 5} v - \frac { 1} { 4}#?

First, multiply each side of the equation #color(red)(60)# to eliminate the fractions while keeping the equation balanced. #color(red)(60)# is the lowest common denominator for the three fractions:

#color(red)(60) xx 1/3 = color(red)(60)(-7/5v - 1/4)#

#color(red)(60)/3 = (color(red)(60) xx -7/5v) - (color(red)(60) xx 1/4)#

#20 = (cancel(color(red)(60))color(red)(12) xx -7/color(red)(cancel(color(black)(5)))v) - color(red)(60)/4#

#20 = (color(red)(12) xx -7v) - 15#

#20 = -84v - 15#

Next, add #color(red)(15)# to each side of the equation to isolate the #v# term while keeping the equation balanced:

#20 + color(red)(15) = -84v - 15 + color(red)(15)#

#35 = -84v - 0#

#35 = -84v#

Now, divide each side of the equation by #color(red)(-84)# to solve for #v# while keeping the equation balanced:

#(35)/color(red)(-84) = (-84v)/color(red)(-84)#

#-(7 xx 5)/color(red)(7 xx 12) = (color(red)(cancel(color(black)(-84)))v)/cancel(color(red)(-84))#

#-(color(red)(cancel(color(black)(7))) xx 5)/color(red)(color(black)(cancel(color(red)(7))) xx 12) = (color(red)(cancel(color(black)(-84)))v)/cancel(color(red)(-84))#

#-5/12 = v#

#v = -5/12#