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))
