Solve the system of equations `2x" "+" "5y" "=" "1` `3x" "+" "2y" "=" "7`

Explanation

The given system oif equation is
2x+5y=1
3x+2y=7
Let `A=[{:(2 " "5),(3 " "2):}], X=[{:(x),(y):}] and B=[{:(1),(7):}]`
Then the given system is AX=B,
Now ,`A=[{:(2 " "5),(3 " "2):}]=(4-15)=-11 ne 0`
Hence the given system has a unique solution
The minors of the elements of |A| are `M_(11)=2,M_(12)=3,`
`M_(21)=5,M_(22)=2`
So, the confactors of the elements of |A|are
`A_(11)=2,A_(12)=-3,`
`A_(21)=-5,A_(22)=2`
`therefore (adj)=[{:(2 "  "-3),(-5 "   "2):}]=[{:(2 "  "-5),(-3 "   "2):}]`
`rArr A^(-1)=1/(|A|)(adjA)=(-1)/(11).[{:(2 " "-5),(-3 "   "2):}]=[{:((-2)/(11) " "5/(11)),(3/(11) " "(-2)/(11)):}]`
`rArr=X=A^(-1)B`
`rArr [{:(x),(y):}]=[{:((-2)/(11) " "5/(11)),(3/(11) " "(-2)/(11)):}][{:(1),(7):}]=[{:(x),(y):}]=[{:((-2)/(11)+35/(11)),(3/(11)-(14)/(11)):}]=[{:(3),(-1):}]`
`rArrx=3 and y=-1`
Hence ,x=3 and y=1