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Tentukan penyelesaian sistem persamaan berikut den...

Tentukan penyelesaian sistem persamaan berikut dengan menggunakan metode determinan matriks!

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell 3 x plus y minus 2 z end cell equals 5 row cell 2 x plus 3 y plus z end cell equals 15 row cell 2 x minus y plus 3 z end cell equals 7 end table end style

Jawaban:

Bentuk matriks dari sistem persamaan linear tersebut adalah sebagai berikut.

begin mathsize 14px style open parentheses table row 3 1 cell negative 2 end cell row 2 3 1 row 2 cell negative 1 end cell 3 end table close parentheses open parentheses table row x row y row z end table close parentheses equals open parentheses table row 5 row 15 row 7 end table close parentheses end style

Berdasarkan konsep determinan metode sarrus diperoleh nilai-nilai berikut.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row D equals cell open vertical bar table row 3 1 cell negative 2 end cell row 2 3 1 row 2 cell negative 1 end cell 3 end table close vertical bar table row 3 1 row 2 3 row 2 cell negative 1 end cell end table end cell row blank equals cell 3 times 3 times 3 plus 1 times 1 times 2 plus open parentheses negative 2 close parentheses times 2 times open parentheses negative 1 close parentheses end cell row blank blank cell negative open parentheses open parentheses negative 2 close parentheses times 3 times 2 close parentheses minus open parentheses 3 times 1 times open parentheses negative 1 close parentheses close parentheses minus open parentheses 1 times 2 times 3 close parentheses end cell row blank equals cell 27 plus 2 plus 4 minus open parentheses negative 12 close parentheses minus open parentheses negative 3 close parentheses minus 6 end cell row blank equals 42 end table end style

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell D subscript x end cell equals cell open vertical bar table row 5 1 cell negative 2 end cell row 15 3 1 row 7 cell negative 1 end cell 3 end table close vertical bar table row 5 1 row 15 3 row 7 cell negative 1 end cell end table end cell row blank equals cell 5 times 3 times 3 plus 1 times 1 times 7 plus open parentheses negative 2 close parentheses times 15 times open parentheses negative 1 close parentheses end cell row blank blank cell negative open parentheses open parentheses negative 2 close parentheses times 3 times 7 close parentheses minus open parentheses 5 times 1 times open parentheses negative 1 close parentheses close parentheses minus open parentheses 1 times 15 times 3 close parentheses end cell row blank equals cell 45 plus 7 plus 30 minus open parentheses negative 42 close parentheses minus open parentheses negative 5 close parentheses minus 45 end cell row blank equals 84 end table end style

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell D subscript y end cell equals cell open vertical bar table row 3 5 cell negative 2 end cell row 2 15 1 row 2 7 3 end table close vertical bar table row 3 5 row 2 15 row 2 7 end table end cell row blank equals cell 3 times 15 times 3 plus 5 times 1 times 2 plus open parentheses negative 2 close parentheses times 2 times 7 end cell row blank blank cell negative open parentheses open parentheses negative 2 close parentheses times 15 times 2 close parentheses minus open parentheses 3 times 1 times 7 close parentheses minus open parentheses 5 times 2 times 3 close parentheses end cell row blank equals cell 135 plus 10 minus 28 minus open parentheses negative 60 close parentheses minus 21 minus 30 end cell row blank equals 126 end table end style

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell D subscript z end cell equals cell open vertical bar table row 3 1 5 row 2 3 15 row 2 cell negative 1 end cell 7 end table close vertical bar table row 3 1 row 2 3 row 2 cell negative 1 end cell end table end cell row blank equals cell 3 times 3 times 7 plus 1 times 15 times 2 plus 5 times 2 times open parentheses negative 1 close parentheses end cell row blank blank cell negative open parentheses 5 times 3 times 2 close parentheses minus open parentheses 3 times 15 times open parentheses negative 1 close parentheses close parentheses minus open parentheses 1 times 2 times 7 close parentheses end cell row blank equals cell 63 plus 30 minus 10 minus 30 plus 45 minus 14 end cell row blank equals 84 end table end style

Nilai begin mathsize 14px style x comma space y comma space z end style dapat ditentukan sebagai berikut.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row x equals cell D subscript x over D equals 84 over 42 equals 2 end cell row y equals cell D subscript y over D equals 126 over 42 equals 3 end cell row z equals cell D subscript z over D equals 84 over 42 equals 2 end cell end table end style

Dengan demikian, penyelesaian sistem persamaan linear adalah begin mathsize 14px style x equals 2 comma space y equals 3 comma space text dan end text space z equals 2 end style 

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