Respuesta :
Answer:
$4500
Step-by-step explanation:
The amount a has is 1/3 total of b, c, and d
[tex]a=1/3(b+c+d)[/tex] eq. 1
The amount b has is 1/4 total of a, c, and d
[tex]b=1/4(a+c+d)[/tex] eq. 2
The amount c has is 1/5 total of a, b, and d
[tex]c=1/5(a+b+d)[/tex] eq. 3
[tex]d=1725[/tex]
substitute the value of [tex]d=1725[/tex] in above equations
[tex]3a=b+c+1725[/tex]
[tex]4b=a+c+1725[/tex]
[tex]5c=a+b+1725[/tex]
Rearrange the equations to form matrices
[tex]3a-b-c=1725[/tex]
[tex]-a+4b-c=1725[/tex]
[tex]-a-b+5c=1725 [/tex]
[tex]X=\left[\begin{array}{c}a&b&c\\\end{array}\right][/tex]
[tex]A=\left[\begin{array}{ccc}3&-1&-1\\-1&4&-1\\-1&-1&5\end{array}\right][/tex]
[tex]B=\left[\begin{array}{c}1725&1725&1725\\\end{array}\right][/tex]
As we know
[tex]AX=B[/tex]
To find the values of matrix X, we take inverse of matrix A and multiply it with matrix B
[tex]X=A^{-1} B[/tex]
[tex]\left[\begin{array}{c}a&b&c\\\end{array}\right]=\left[\begin{array}{ccc}3&-1&-1\\-1&4&-1\\-1&-1&5\end{array}\right]^{-1} \left[\begin{array}{c}1725&1725&1725\\\end{array}\right][/tex]
Solving using calculator yield the following results
[tex]X=\left[\begin{array}{c}1125&900&750\\\end{array}\right][/tex]
so,
[tex]a=1125\\b=900\\c=750\\d=1725\\[/tex]
Finally, altogether they have
[tex]1125+900+750+1725=4500[/tex]
Verification:
Lets verify if have got the right answer!
Substitute the amount of b, c, and d in eq. 1
[tex]a=1/3(b+c+d)[/tex]
[tex]a=1/3(900+750+1725)[/tex]
[tex]a=1/3(3375)[/tex]
[tex]a=1125[/tex] (proved)
Substitute the amount of a, c, and d in eq. 2
[tex]b=1/4(a+c+d)[/tex]
[tex]b=1/4(1125+750+1725)[/tex]
[tex]b=1/4(3600)[/tex]
[tex]b=900[/tex] (proved)
Substitute the amount of a, b, and d in eq. 3
[tex]c=1/5(a+b+d)[/tex]
[tex]c=1/5(1125+900+1725)[/tex]
[tex]c=1/5(3750)[/tex]
[tex]c=750[/tex] (proved)
