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Tentukan penyelesaian sistem persamaan berikut dengan menggunakan metode determinan matriks!

Pertanyaan

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 5 x minus y plus z end cell equals 6 row cell x plus 3 y minus z end cell equals 4 row cell x plus y minus 2 z end cell equals cell negative 3 end cell end table end style

Pembahasan Soal:

Bentuk matriks dari sistem persamaan linear tersebut adalah sebagai berikut.

begin mathsize 14px style open parentheses table row 5 cell negative 1 end cell 1 row 1 3 cell negative 1 end cell row 1 1 cell negative 2 end cell end table close parentheses open parentheses table row x row y row z end table close parentheses equals open parentheses table row 6 row 4 row cell negative 3 end cell 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 5 cell negative 1 end cell 1 row 1 3 cell negative 1 end cell row 1 1 cell negative 2 end cell end table close vertical bar table row 5 cell negative 1 end cell row 1 3 row 1 1 end table end cell row blank equals cell 5 times 3 times open parentheses negative 2 close parentheses plus open parentheses negative 1 close parentheses open parentheses negative 1 close parentheses open parentheses 1 close parentheses plus 1 times 1 times 1 end cell row blank blank cell negative open parentheses 1 times 3 times 1 close parentheses minus open parentheses 5 open parentheses negative 1 close parentheses times 1 close parentheses minus open parentheses open parentheses negative 1 close parentheses times 1 open parentheses negative 2 close parentheses close parentheses end cell row blank equals cell negative 30 plus 1 plus 1 minus 3 plus 5 minus 2 end cell row blank equals cell negative 28 end cell 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 6 cell negative 1 end cell 1 row 4 3 cell negative 1 end cell row cell negative 3 end cell 1 cell negative 2 end cell end table close vertical bar table row 6 cell negative 1 end cell row 4 3 row cell negative 3 end cell 1 end table end cell row blank equals cell 6 times 3 times open parentheses negative 2 close parentheses plus open parentheses negative 1 close parentheses open parentheses negative 1 close parentheses open parentheses negative 3 close parentheses plus 1 times 4 times 1 end cell row blank blank cell negative open parentheses 1 times 3 open parentheses negative 3 close parentheses close parentheses minus open parentheses 6 times open parentheses negative 1 close parentheses times 1 close parentheses minus open parentheses open parentheses negative 1 close parentheses times 4 open parentheses negative 2 close parentheses close parentheses end cell row blank equals cell negative 36 minus 3 plus 4 plus 9 plus 6 minus 8 end cell row blank equals cell negative 28 end cell 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 5 6 1 row 1 4 cell negative 1 end cell row 1 cell negative 3 end cell cell negative 2 end cell end table close vertical bar table row 5 6 row 1 4 row 1 cell negative 3 end cell end table end cell row blank equals cell 5 times 4 times open parentheses negative 2 close parentheses plus 6 times open parentheses negative 1 close parentheses times 1 plus 1 times 1 times open parentheses negative 3 close parentheses end cell row blank blank cell negative open parentheses 1 times 4 times 1 close parentheses minus open parentheses 5 times open parentheses negative 1 close parentheses times open parentheses negative 3 close parentheses close parentheses minus open parentheses 6 times 1 times open parentheses negative 2 close parentheses close parentheses end cell row blank equals cell negative 40 minus 6 minus 3 minus 4 minus 15 plus 12 end cell row blank equals cell negative 56 end cell 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 5 cell negative 1 end cell 6 row 1 3 4 row 1 1 cell negative 3 end cell end table close vertical bar table row 5 cell negative 1 end cell row 1 3 row 1 1 end table end cell row blank equals cell 5 times 3 times open parentheses negative 3 close parentheses plus open parentheses negative 1 close parentheses times 4 times 1 plus 6 times 1 times 1 end cell row blank blank cell negative open parentheses 6 times 3 times 1 close parentheses minus open parentheses 5 times 4 times 1 close parentheses minus open parentheses open parentheses negative 1 close parentheses times 1 times open parentheses negative 3 close parentheses close parentheses end cell row blank equals cell negative 45 minus 4 plus 6 minus 18 minus 20 minus 3 end cell row blank equals cell negative 84 end cell 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 fraction numerator negative 28 over denominator negative 28 end fraction equals 1 end cell row y equals cell D subscript y over D equals fraction numerator negative 56 over denominator negative 28 end fraction equals 2 end cell row z equals cell D subscript z over D equals fraction numerator negative 84 over denominator negative 28 end fraction equals 3 end cell end table end style

Dengan demikian, penyelesaian sistem persamaan linear adalah begin mathsize 14px style x equals 1 comma space y equals 2 comma space text dan end text space z equals 3 end style

Pembahasan terverifikasi oleh Roboguru

Terakhir diupdate 02 April 2021

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