Put the eqns in matrix form
#x-2y=2#
#3x-5y=9#
become
#((1,-2),(3,-5))##((x),(y))=((2),(9))#
we therefore have
#M((x),(y))=((2),(9))#
by premultiplying both sides by #M^(-1)#
#M^(-1)M((x),(y))=M^(-1)((2),(9))#
#I((x),(y))=M^(-1)((2),(9))#
where #I=((1,0),(0,1))#
#:.((x),(y))=M^(-1)((2),(9))#
so we need to find the inverse of the matrix.
#M=((1,-2),(3,-5))#
for any # 2xx2 # mx
#M=((a,b),(c,d))#
#M^(-1)=1/Delta((d,-b),(-c,a))" providing "DeltaM!=0#
find its determinant
#Delta M=|(1,-2),(3,-5)|#
#DeltaM=1xx-5-(3xx-2)=-5--6=1#
#because Delta M!=0 " the inverse exists"#
#:.M^(-1)=1/1((-5,2),(-3,1))=((-5,2),(-3,1))#
so#((x),(y))=M^(-1)((2),(9))=((-5,2),(-3,1))((2),(9))#
#((x),(y))=((-5xx2+2xx9),(-3xx2+1xx9))#
#((x),(y))=((-10+18),(-6+9))#
#((x),(y))=((-10+18),(-6+9))#
#((x),(y))=((8),(3))#