We are given:
\(\displaystyle\frac{{{7}{y}+{3}}}{{2}}={\left(\frac{{{8}{y}+{1}}}{{6}}\right)}+{23}\)

Multiply both sides by the LCD which is 6: \(\displaystyle\frac{{{7}{y}+{3}}}{{2}}{\left({6}\right)}={\left({\left(\frac{{{8}{y}+{1}}}{{6}}\right)}+{23}\right)}{\left({6}\right)}\)

21y+9=(8y+1)+138

21y+9=8y+139

Subtract 9 from both sides: 21y=8y+130

Subtract 8y from both sides: 13y=130

Divie both sides by 13: y=10

Multiply both sides by the LCD which is 6: \(\displaystyle\frac{{{7}{y}+{3}}}{{2}}{\left({6}\right)}={\left({\left(\frac{{{8}{y}+{1}}}{{6}}\right)}+{23}\right)}{\left({6}\right)}\)

21y+9=(8y+1)+138

21y+9=8y+139

Subtract 9 from both sides: 21y=8y+130

Subtract 8y from both sides: 13y=130

Divie both sides by 13: y=10