Mr. Vogel divided $25,000 between his daughter and his son. Together they both received $25,000. However, the daughter received $5,000 less than the son. Write a system of equations that represents this situation using matrices. How much did each child receive?
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1 Answer

The system in matrix form is:

( 1  1 )( s )=( 25000 )

( 1 -1 )( d )=( 5000 )

Representing the equations:

s+d=25000

s-d=5000 where s=son’s share, d=daughter’s share.

Let matrix A=

( 1   1 )

( 1  -1 )

B=

( 25000 )

(  5000  )

and X=

( s )

( d )

So we have AX=B.

Therefore X=A⁻¹B.

A⁻¹=

½( 1  1 )

   ( 1 -1 )

So

( s ) = ½( 1  1 )( 25000 )

( d )       ( 1 -1 )(  5000  ) from which s=½(25000+5000)=15000 

and d=½(25000-5000)=10000.

The son gets $15000 and the daughter $10000.

[To find the inverse of A we solve A×A⁻¹=I.

( 1  1 )( a b )=( 1 0 )

( 1 -1 )( c d )  ( 0 1 )

From this, a+c=1, a-c=0, b+d=0, b-d=1.

So a=c=½, d=-b, b=½, d=-½. So the inverse is:

½( 1  1 )

   ( 1 -1 ) or

-½( -1 -1)

    ( -1 1 ) where the scalar ½ or -½ is a common factor for the elements and can be taken outside the matrix.]

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