Solve system of linear equations, using matrix method,
`5x + 2y = 4," "``7x + 3 y = 5`

Explanation

The system of equation is

`5x + 2y = 4`

`7x + 3 y = 5`


Writing equation as AX=B

`[[5,2],[7,3]][[x],[y]]=[[4],[5]]`

Here,
`A=[[5,2],[7,3]]`, `X=[[x],[y]]` &`B=[[4],[5]]`


Calculating `|A|`

`|A|=|[5,2],[7,3]|`
` " " " =5(3)-7(2)=15-14`
` " " " =1`
`|A|ne0`

Since `|A|ne0`
`therefore` System of equation is consistent & has a unique solution.

Now,
`AX=B`
Solution is `X=A^(-1)B`


Calculating `A^(-1)`

`A^(-1)=1/|A| adj(A)`

` adj(A)=[[3,-2],[-7,5]]`

`A^(-1)=1/|A| adj(A)`
Putting values

` A^(-1)=1/1[[3,-2],[-7,5]]=[[3,-2],[-7,5]]`

Now,
`X=A^(-1)B`

`[[x],[y]]=[[3,-2],[-7,5]][[4],[5]]`

`[[x],[y]]=[[3(4)+(-2)5],[-7(4)+(5)5]]`

`[[x],[y]]=[[2],[-3]]`

Hence, `x=2` & `y=-3`