Question 136242
I have to assume that you meant:  {{{(y-3)/5=(y+1)/7}}}, because what you wrote:  {{{y-3/5=y+1/7}}} has no solution.


{{{(y-3)/5=(y+1)/7}}}


7 and 5 have no common factors so the LCD is 7*5=35, so:
{{{7(y-3)/35=5(y+1)/35}}}


Add the additive inverse of the right side to both sides:
{{{7(y-3)/35-5(y+1)/35=0}}}


Since both fractions have the same denominator, add the numerators directly:
{{{(7(y-3)-5(y+1))/35=0}}}


Distribute and collect like terms:
{{{(2y-26)/35=0}}}


We know that {{{a/b=0}}} if and only if {{{a=0}}} and {{{b<>0}}}.  The denominator is not a problem because clearly {{{35<>0}}}, so all we need do is set the numerator equal to zero and solve:
{{{2y-26=0}}}
{{{2y=26}}}
{{{y=13}}}


Check:
{{{(13-3)/5=(13+1)/7}}}
{{{10/5=14/7}}}
{{{2=2}}}, Answer checks.