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Pertanyaan

Tentukan nilai x dan y dari persamaan linier 2  variabel dengan menggunakan:

a. matriks

b. aturan cramer

c. eliminasi

d. subsitusi

e. gabungan

 open curly brackets table attributes columnalign left end attributes row cell negative 8 x minus 2 y equals 8 end cell row cell negative 3 x plus y equals negative 2 end cell end table close

M. Nasrullah

Master Teacher

Mahasiswa/Alumni Universitas Negeri Makassar

Jawaban terverifikasi

Pembahasan

Perhatikan perhitungan berikut:

-bentuk matriks:

A X equals B rightwards arrow X equals A to the power of negative 1 end exponent times B 

Pada soal diketahui:

open curly brackets table attributes columnalign left end attributes row cell negative 8 x minus 2 y equals 8 end cell row cell negative 3 x plus y equals negative 2 end cell end table close rightwards arrow open parentheses table row cell negative 8 end cell cell negative 2 end cell row cell negative 3 end cell 1 end table close parentheses open parentheses table row x row y end table close parentheses equals open parentheses table row 8 row cell negative 2 end cell end table close parentheses

sehingga:

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses table row x row y end table close parentheses end cell equals cell fraction numerator 1 over denominator negative 8 minus 6 end fraction open parentheses table row 1 2 row 3 cell negative 8 end cell end table close parentheses open parentheses table row 8 row cell negative 2 end cell end table close parentheses end cell row blank equals cell fraction numerator 1 over denominator negative 14 end fraction open parentheses table row 1 2 row 3 cell negative 8 end cell end table close parentheses open parentheses table row 8 row cell negative 2 end cell end table close parentheses end cell row blank equals cell fraction numerator 1 over denominator negative 14 end fraction open parentheses table row cell 8 minus 4 end cell row cell 24 plus 16 end cell end table close parentheses end cell row blank equals cell fraction numerator 1 over denominator negative 14 end fraction open parentheses table row 4 row 40 end table close parentheses end cell row blank equals cell open parentheses table row cell bevelled fraction numerator negative 4 over denominator 14 end fraction end cell row cell bevelled fraction numerator negative 40 over denominator 14 end fraction end cell end table close parentheses end cell row blank equals cell open parentheses table row cell bevelled fraction numerator negative 2 over denominator 7 end fraction end cell row cell bevelled fraction numerator negative 20 over denominator 7 end fraction end cell end table close parentheses end cell end table 

Jadi, nilai x dan y masing-masing adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator negative 2 over denominator 7 end fraction end cell end table dan table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator negative 20 over denominator 7 end fraction end cell end table.

b. aturan cramer

open curly brackets table attributes columnalign left end attributes row cell negative 8 x minus 2 y equals 8 end cell row cell negative 3 x plus y equals negative 2 end cell end table close rightwards arrow open parentheses table row cell negative 8 end cell cell negative 2 end cell row cell negative 3 end cell 1 end table close parentheses open parentheses table row x row y end table close parentheses equals open parentheses table row 8 row cell negative 2 end cell end table close parentheses
table attributes columnalign right center left columnspacing 0px end attributes row D equals cell open vertical bar table row cell negative 8 end cell cell negative 2 end cell row cell negative 3 end cell 1 end table close vertical bar end cell row blank equals cell negative 8 minus 6 end cell row blank equals cell negative 14 end cell end table 

table attributes columnalign right center left columnspacing 0px end attributes row cell D subscript x end cell equals cell open vertical bar table row 8 cell negative 2 end cell row cell negative 2 end cell 1 end table close vertical bar end cell row blank equals cell 8 minus 4 end cell row blank equals 4 end table

table attributes columnalign right center left columnspacing 0px end attributes row cell D subscript y end cell equals cell open vertical bar table row cell negative 8 end cell 8 row cell negative 3 end cell cell negative 2 end cell end table close vertical bar end cell row blank equals cell 16 plus 24 end cell row blank equals 40 end table

sehingga,

x equals D subscript x over D equals fraction numerator 4 over denominator negative 14 end fraction equals fraction numerator negative 2 over denominator 7 end fraction y equals D subscript y over D equals fraction numerator 40 over denominator negative 14 end fraction equals fraction numerator negative 20 over denominator 7 end fraction 

Jadi, nilai x dan y masing-masing adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator negative 2 over denominator 7 end fraction end cell end table dan table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator negative 20 over denominator 7 end fraction end cell end table.

c. eliminasi

open curly brackets table attributes columnalign left end attributes row cell negative 8 x minus 2 y equals 8 end cell row cell negative 3 x plus y equals negative 2 end cell end table close

Langsung kita eliminasi  x pada kedua persamaan tersebut:

 table row cell negative 8 x minus 2 y equals 8 end cell cell open vertical bar cross times 3 close vertical bar end cell row cell negative 3 x plus y equals negative 2 end cell cell open vertical bar cross times 8 close vertical bar end cell row blank blank row blank blank row blank blank row blank blank row blank blank end table table row cell negative 24 x minus 6 y equals 24 end cell row cell negative 24 x plus 8 equals negative 16 end cell row cell negative 14 y equals 40 end cell row cell y equals fraction numerator 40 over denominator negative 14 end fraction end cell row cell y equals fraction numerator negative 20 over denominator 7 end fraction end cell end table minus 

eliminasi  y pada kedua persamaan tersebut:

table row cell negative 8 x minus 2 y equals 8 end cell cell open vertical bar cross times 1 close vertical bar end cell row cell negative 3 x plus y equals negative 2 end cell cell open vertical bar cross times 2 close vertical bar end cell row blank blank row blank blank row blank blank row blank blank row blank blank end table table row cell negative 8 x minus 2 y equals 8 end cell row cell negative 6 x plus 2 x equals negative 4 end cell row cell negative 14 x equals 4 end cell row cell x equals fraction numerator 4 over denominator negative 14 end fraction end cell row cell x equals fraction numerator negative 2 over denominator 7 end fraction end cell end table plus

Jadi, nilai x dan y masing-masing adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator negative 2 over denominator 7 end fraction end cell end table dan table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator negative 20 over denominator 7 end fraction end cell end table.

d. subsitusi

table row cell negative 8 x minus 2 y equals 8.... open parentheses 1 close parentheses end cell row blank row cell negative 3 x plus y equals negative 2 end cell row cell y equals negative 2 plus 3 x.... open parentheses 2 close parentheses end cell end table

Subtitusi persamaan 2 ke persamaan 1

table attributes columnalign right center left columnspacing 0px end attributes row cell negative 8 x minus 2 y end cell equals 8 row cell negative 8 x minus 2 open parentheses negative 2 plus 3 x close parentheses end cell equals 8 row cell negative 8 x plus 4 minus 6 x end cell equals 8 row cell negative 14 x end cell equals cell 8 minus 4 end cell row x equals cell fraction numerator 4 over denominator negative 14 end fraction end cell row x equals cell fraction numerator negative 2 over denominator 7 end fraction end cell end table

Subtitusi nilai x ke persamaan 2

table attributes columnalign right center left columnspacing 0px end attributes row y equals cell negative 2 plus 3 x end cell row y equals cell negative 2 plus 3 open parentheses fraction numerator negative 2 over denominator 7 end fraction close parentheses end cell row blank equals cell negative 2 minus 6 over 7 end cell row blank equals cell fraction numerator negative 14 minus 6 over denominator 7 end fraction end cell row blank equals cell fraction numerator negative 20 over denominator 7 end fraction end cell end table 

Jadi, nilai x dan y masing-masing adalah  table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator negative 2 over denominator 7 end fraction end cell end table dan table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator negative 20 over denominator 7 end fraction end cell end table.

e. gabungan (eliminasi subtitusi)

eliminasi  y pada kedua persamaan tersebut:

table row cell negative 8 x minus 2 y equals 8 end cell cell open vertical bar cross times 1 close vertical bar end cell row cell negative 3 x plus y equals negative 2 end cell cell open vertical bar cross times 2 close vertical bar end cell row blank blank row blank blank row blank blank row blank blank row blank blank end table table row cell negative 8 x minus 2 y equals 8 end cell row cell negative 6 x plus 2 x equals negative 4 end cell row cell negative 14 x equals 4 end cell row cell x equals fraction numerator 4 over denominator negative 14 end fraction end cell row cell x equals fraction numerator negative 2 over denominator 7 end fraction end cell end table plus

Subtitusi nilai x ke persamaan 2

table attributes columnalign right center left columnspacing 0px end attributes row cell negative 3 x plus y end cell equals cell negative 2 end cell row cell negative 3 open parentheses fraction numerator negative 2 over denominator 7 end fraction close parentheses plus y end cell equals cell negative 2 end cell row cell 6 over 7 plus y end cell equals cell negative 2 end cell row y equals cell negative 2 minus fraction numerator negative 6 over denominator 7 end fraction end cell row blank equals cell fraction numerator negative 14 minus 6 over denominator 7 end fraction end cell row blank equals cell fraction numerator negative 20 over denominator 7 end fraction end cell end table

Dengan demikian, nilai x dan y masing-masing adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator negative 2 over denominator 7 end fraction end cell end table dan table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator negative 20 over denominator 7 end fraction end cell end table.

 

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