a x + b y = u {\displaystyle ax+by=u\,}
c x + d y = v {\displaystyle cx+dy=v\,}
a c x + b c y = u c {\displaystyle acx+bcy=uc\,}
a c x + a d y = v a {\displaystyle acx+ady=va\,}
( a c x + b c y ) − ( a c x + a d y ) = u c − v a {\displaystyle (acx+bcy)-(acx+ady)=uc-va\,}
b c y − a d y = u c − v a {\displaystyle bcy-ady=uc-va\,}
y ( b c − a d ) = u c − v a {\displaystyle y(bc-ad)=uc-va\,}
y = u c − v a b c − a d {\displaystyle y={\frac {uc-va}{bc-ad}}}
