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Question

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

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Solution

Given system of equations
5x+2y=4
7x+3y=5
This can be written as
AX=B
where A=[5273],X=[xy],B=[45]

Here, |A|=1514=1
Since, |A|0
Hence, A1 exists and the system has a unique solution given by X=A1B

A1=adjA|A| and adjA=CT

So, we will find the co-factors of each element of A.
C11=(1)1+13=3
C12=(1)1+27=7
C21=(1)2+12=2
C22=(1)2+25=5

So, the co-factor matrix is [3725]

adjA=CT=[3275]

A1=adjA|A|=[3275]

The solution is X=A1B
[xy]=[3275][45]
=[121028+25]
[xy]=[23]
Hence, x=2,y=3

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