A man has two investments, one paying 3% annual interest and the other 4%. The total annual income from the investments is Rp1,870,000. If the interest rates were interchanged, the total annual income would be Rp1,980,000. Find the amount of each investment.

Pertanyaan

A man has two investments, one paying begin mathsize 14px style 3 percent sign end style annual interest and the other begin mathsize 14px style 4 percent sign end style. The total annual income from the investments is begin mathsize 14px style Rp 1 comma 870 comma 000 end style. If the interest rates were interchanged, the total annual income would be begin mathsize 14px style Rp 1 comma 980 comma 000 end style. Find the amount of each investment.

H. Endah

Master Teacher

Mahasiswa/Alumni Universitas Negeri Yogyakarta

Jawaban terverifikasi

Jawaban

 jumlah setiap investasi adalah Rp33.000.000,00 dan Rp22.000.000,00.

Pembahasan

Menentukan penyelesaian SPLDV dengan Metode Invers Matriks adalah sebagai berikut:

table attributes columnalign right center left columnspacing 0px end attributes row blank blank a end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank b end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank y end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank e end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank d end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank y end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank f end table  

Diubah ke dalam bentuk matriks didapatkan:

open parentheses table row a b row c d end table close parentheses open parentheses table row x row y end table close parentheses equals open parentheses table row e row f end table close parentheses space horizontal ellipsis space open parentheses 1 close parentheses  

Persamaan open parentheses 1 close parentheses dapat ditulis sebagai A X equals B, dengan:

A equals open parentheses table row a b row c d end table close parentheses comma space X equals open parentheses table row x row y end table close parentheses comma space dan space B equals open parentheses table row e row f end table close parentheses  

Penentuan X dapat dilakukan dengan sifat matriks, yaitu:

table attributes columnalign right center left columnspacing 0px end attributes row cell A X end cell equals B row X equals cell A to the power of negative 1 end exponent B end cell end table   

Diketahui A equals open parentheses table row a b row c d end table close parentheses, maka inversnya adalah:

A to the power of negative 1 end exponent equals fraction numerator 1 over denominator a d minus b c end fraction open parentheses table row d cell negative b end cell row cell negative c end cell a end table close parentheses 

Misalkan investasi A adalah x dan investasi B adalah y, maka didapatkan persamaan:

table row cell 3 over 100 x plus 4 over 100 y equals 1.870.000 end cell rightwards double arrow cell 3 x plus 4 y equals 187.000.000 end cell row cell 4 over 100 x plus 3 over 100 y equals 1.980.000 end cell rightwards double arrow cell 4 x plus 3 y equals 198.000.000 end cell end table    

Dari 2 persamaan di atas diubah ke dalam matriks dan didapatkan:

 open parentheses table row 3 4 row 4 3 end table close parentheses open parentheses table row x row y end table close parentheses equals open parentheses table row cell 187.000.000 end cell row cell 198.000.000 end cell end table close parentheses 

Misalkan A equals open parentheses table row 3 4 row 4 3 end table close parentheses comma space X equals open parentheses table row x row y end table close parentheses comma space dan space B equals open parentheses table row cell 187.000.000 end cell row cell 198.000.000 end cell end table close parentheses, maka matriks X didapatkan:

table attributes columnalign right center left columnspacing 0px end attributes row cell A X end cell equals B row X equals cell A to the power of negative 1 end exponent B end cell row cell open parentheses table row x row y end table close parentheses end cell equals cell open parentheses table row 3 4 row 4 3 end table close parentheses to the power of negative 1 end exponent open parentheses table row cell 187.000.000 end cell row cell 198.000.000 end cell end table close parentheses end cell row cell open parentheses table row x row y end table close parentheses end cell equals cell fraction numerator 1 over denominator open parentheses 3 close parentheses open parentheses 3 close parentheses minus open parentheses 4 close parentheses open parentheses 4 close parentheses end fraction open parentheses table row 3 cell negative 4 end cell row cell negative 4 end cell 3 end table close parentheses open parentheses table row cell 187.000.000 end cell row cell 198.000.000 end cell end table close parentheses end cell row cell open parentheses table row x row y end table close parentheses end cell equals cell fraction numerator 1 over denominator 9 minus 16 end fraction open parentheses table row cell negative 231.000.000 end cell row cell negative 154.000.000 end cell end table close parentheses end cell row cell open parentheses table row x row y end table close parentheses end cell equals cell fraction numerator 1 over denominator negative 7 end fraction open parentheses table row cell negative 231.000.000 end cell row cell negative 154.000.000 end cell end table close parentheses end cell row cell open parentheses table row x row y end table close parentheses end cell equals cell open parentheses table row cell 33.000.000 end cell row cell 22.000.000 end cell end table close parentheses end cell end table   

Maka investasi A adalah Rp33.000.000,00 dan investasi B adalah Rp22.000.000,00.

Jadi, jumlah setiap investasi adalah Rp33.000.000,00 dan Rp22.000.000,00.

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