${\displaystyle ax+by=u\,}$

${\displaystyle cx+dy=v\,}$

${\displaystyle acx+bcy=uc\,}$

${\displaystyle acx+ady=va\,}$

${\displaystyle (acx+bcy)-(acx+ady)=uc-va\,}$

${\displaystyle bcy-ady=uc-va\,}$

${\displaystyle y(bc-ad)=uc-va\,}$

${\displaystyle y={\frac {uc-va}{bc-ad}}}$