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Perhatikan gambar berikut!

Nilai maksimum fungsi objektif begin mathsize 14px style f left parenthesis x comma y right parenthesis equals 2 x plus y end style dari daerah penyelesaian di atas adalah ....

  1. begin mathsize 14px style negative 9 over 2 end style 

  2. begin mathsize 14px style negative 24 over 7 end style 

  3. 0

  4. begin mathsize 14px style 24 over 7 end style 

  5. begin mathsize 14px style 9 over 2 end style 

H. Nufus

Master Teacher

Mahasiswa/Alumni Universitas Negeri Surabaya

Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah C.

Pembahasan

Pertama, kita cari masing-masing persamaan garis di atas.

Diketahui garis berwarna merah melalui titik (-2,0) dan (0,-3) maka persamaan garisnya

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator y minus y subscript 1 over denominator y subscript 2 minus y subscript 1 end fraction end cell equals cell fraction numerator x minus x subscript 1 over denominator x subscript 2 minus x subscript 1 end fraction end cell row cell fraction numerator y minus 0 over denominator negative 3 minus 0 end fraction end cell equals cell fraction numerator x minus open parentheses negative 2 close parentheses over denominator 0 minus open parentheses negative 2 close parentheses end fraction end cell row cell fraction numerator y over denominator negative 3 end fraction end cell equals cell fraction numerator x plus 2 over denominator 2 end fraction end cell row cell 2 y end cell equals cell negative 3 x minus 6 end cell row cell 2 y plus 3 x plus 6 end cell equals 0 end table end style 

Selanjutnya, garis berwarna biru melalui titik (-1,-2) dan (1,2) maka persamaan garisnya

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator y minus y subscript 1 over denominator y subscript 2 minus y subscript 1 end fraction end cell equals cell fraction numerator x minus x subscript 1 over denominator x subscript 2 minus x subscript 1 end fraction end cell row cell fraction numerator y minus open parentheses negative 2 close parentheses over denominator 2 minus open parentheses negative 2 close parentheses end fraction end cell equals cell fraction numerator x minus open parentheses negative 1 close parentheses over denominator 1 minus open parentheses negative 1 close parentheses end fraction end cell row cell fraction numerator y plus 2 over denominator 4 end fraction end cell equals cell fraction numerator x plus 1 over denominator 2 end fraction end cell row cell 2 y plus 4 end cell equals cell 4 x plus 4 end cell row cell 2 y end cell equals cell 4 x end cell row y equals cell 2 x end cell end table end style 

Selanjutnya, garis berwarna hijau melalui titik (-4,2) dan (4,-2) maka persamaan garisnya

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator y minus y subscript 1 over denominator y subscript 2 minus y subscript 1 end fraction end cell equals cell fraction numerator x minus x subscript 1 over denominator x subscript 2 minus x subscript 1 end fraction end cell row cell fraction numerator y minus 2 over denominator negative 2 minus 2 end fraction end cell equals cell fraction numerator x minus open parentheses negative 4 close parentheses over denominator 4 minus open parentheses negative 4 close parentheses end fraction end cell row cell fraction numerator y minus 2 over denominator negative 4 end fraction end cell equals cell fraction numerator x plus 4 over denominator 8 end fraction end cell row cell 8 y minus 16 end cell equals cell negative 4 x minus 16 end cell row cell 8 y end cell equals cell negative 4 x end cell row cell 2 y end cell equals cell negative x end cell end table end style 

Lalu, kita cari titik potong antar dua garis di atas.

  • Titik potong antara garis 2y + 3x + 6 = 0 dan y = 2x

    begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell 2 y plus 3 x plus 6 end cell equals 0 row cell 2 left parenthesis 2 x right parenthesis plus 3 x plus 6 end cell equals 0 row cell 4 x plus 3 x end cell equals cell negative 6 end cell row cell 7 x end cell equals cell negative 6 end cell row x equals cell negative 6 over 7 end cell row blank blank blank row blank blank blank row y equals cell 2 x end cell row y equals cell 2 open parentheses negative 6 over 7 close parentheses end cell row y equals cell negative 12 over 7 end cell end table end style

Titik potong antara garis 2y + 3x + 6 = 0 dan y = 2x adalah begin mathsize 14px style open parentheses negative 6 over 7 comma negative 12 over 7 close parentheses end style.

  • Titik potong antara garis 2y + 3x + 6 = 0 dan 2y = -x

    begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell 2 y plus 3 x plus 6 end cell equals 0 row cell 2 y plus 3 left parenthesis negative 2 y right parenthesis plus 6 end cell equals 0 row cell 2 y minus 6 y end cell equals cell negative 6 end cell row cell negative 4 y end cell equals cell negative 6 end cell row y equals cell 6 over 4 end cell row y equals cell 3 over 2 end cell row blank blank blank row cell 2 y end cell equals cell negative x end cell row x equals cell negative 2 open parentheses 3 over 2 close parentheses end cell row x equals cell negative 3 end cell end table end style

Titik potong antara garis 2y + 3x + 6 = 0 dan 2y = -x adalah begin mathsize 14px style open parentheses negative 3 comma 3 over 2 close parentheses end style.
 

Perhatikan pada gambar bahwa titik potong antara garis 2y = -x dan y = 2x adalah (0,0).

Kita subtitusikan tiap titik Error converting from MathML to accessible text., (0,0) ke fungsi objektif begin mathsize 14px style f left parenthesis x comma y right parenthesis equals 2 x plus y end style.

  • untuk titik begin mathsize 14px style open parentheses negative 6 over 7 comma negative 12 over 7 close parentheses end style

    begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell f open parentheses negative 6 over 7 comma negative 12 over 7 close parentheses end cell equals cell 2 open parentheses negative 6 over 7 close parentheses plus open parentheses negative 12 over 7 close parentheses end cell row blank equals cell negative 12 over 7 minus 12 over 7 end cell row blank equals cell negative 24 over 7 end cell end table end style
  • untuk titik begin mathsize 14px style open parentheses negative 3 comma 3 over 2 close parentheses end style

    table attributes columnalign right center left columnspacing 0px end attributes row cell size 14px f begin mathsize 14px style open parentheses negative 3 comma 3 over 2 close parentheses end style end cell size 14px equals cell size 14px 2 begin mathsize 14px style left parenthesis negative 3 right parenthesis end style size 14px plus size 14px 3 over size 14px 2 end cell row blank size 14px equals cell size 14px minus size 14px 6 size 14px plus size 14px 3 over size 14px 2 end cell row blank size 14px equals cell size 14px minus size 14px 9 over size 14px 2 end cell end table
  • untuk titik begin mathsize 14px style left parenthesis 0 comma 0 right parenthesis end style

    begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell f left parenthesis 0 comma 0 right parenthesis end cell equals cell 2 left parenthesis 0 right parenthesis plus 0 end cell row blank equals cell 0 plus 0 end cell row blank equals 0 end table end style

Dengan demikian, nilai maksimumnya adalah 0.
 

Jadi, jawaban yang tepat adalah C.

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Pertanyaan serupa

Perhatikan gambar berikut! Nilai minimum fungsi objektif f(x,y) = 3x - y dari daerah penyelesaian di atas adalah ....

37

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