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Tentukan nilai maksimum dari untuk sistem pertidaksamaan :

Tentukan nilai maksimum dari begin mathsize 14px style f left parenthesis x comma y right parenthesis equals 5 x plus 3 y end style untuk sistem pertidaksamaan :

begin mathsize 14px style x plus y less or equal than 6 2 x plus 3 y less or equal than 15 x greater or equal than 0 semicolon y greater or equal than 0 end style 

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Jawaban:

Diketahui :

begin mathsize 14px style x plus y less or equal than 6 space......... space left parenthesis 1 right parenthesis 2 x plus 3 y less or equal than 15 space.... space left parenthesis 2 right parenthesis x greater or equal than 0 semicolon y greater or equal than 0 end style  

Cari titik potong dari persamaan begin mathsize 14px style bold italic x bold plus bold italic y bold less or equal than bold 6 end style  :

Titik potong sumbu begin mathsize 14px style x end stylebegin mathsize 14px style y equals 0 end style 

begin mathsize 14px style x plus y equals 6 x plus 0 equals 6 x equals 6 end style

begin mathsize 14px style rightwards double arrow left parenthesis 6 comma 0 right parenthesis end style 

Titik potong sumbu begin mathsize 14px style y comma space x equals 0 end style

begin mathsize 14px style x plus y equals 6 0 plus y equals 6 y equals 6 end style 

begin mathsize 14px style rightwards double arrow left parenthesis 0 comma 6 right parenthesis end style 

Cari titik potong dari persamaan begin mathsize 14px style bold 2 bold italic x bold plus bold 3 bold italic y bold less or equal than bold 15 end style :

Titik potong sumbu begin mathsize 14px style x end stylebegin mathsize 14px style y equals 0 end style 

begin mathsize 14px style 2 x plus 3 y equals 15 2 x plus 3 left parenthesis 0 right parenthesis equals 15 2 x plus 0 equals 15 2 x equals 15 x equals 15 over 2 end style  

begin mathsize 14px style rightwards double arrow left parenthesis 15 over 2 comma 0 right parenthesis end style 

Titik potong sumbu begin mathsize 14px style y comma space x equals 0 end style

begin mathsize 14px style 2 x plus 3 y equals 15 2 left parenthesis 0 right parenthesis plus 3 y equals 15 0 plus 3 y equals 15 3 y equals 15 y equals 15 over 3 y equals 5 end style 

begin mathsize 14px style rightwards double arrow left parenthesis 0 comma 5 right parenthesis end style 

Eliminasi persamaan (1) dan (2)

begin mathsize 14px style space space x plus space space space y equals 6 space space space open vertical bar cross times 2 space close vertical bar space 2 x plus 2 y equals 12 bottom enclose 2 x plus 3 y equals 15 space space open vertical bar cross times 1 space close vertical bar space 2 x plus 3 y equals 15 space end enclose space space _ space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space minus y equals negative 3 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space y equals 3 end style 

Substitusi begin mathsize 14px style y equals 3 end style ke persamaanbegin mathsize 14px style space x plus y equals 6 end style 

begin mathsize 14px style x plus y equals 6 x plus left parenthesis 3 right parenthesis equals 6 x equals 6 minus 3 x equals 2 end style 

Maka titik potong persamaan (1) dan (2) adalah :

begin mathsize 14px style rightwards double arrow left parenthesis 2 comma 3 right parenthesis end style 

Substitusikan semua titik yang diperoleh pada persamaan undefined 

begin mathsize 14px style rightwards double arrow left parenthesis 6 comma 0 right parenthesis end style 

begin mathsize 14px style f left parenthesis x comma y right parenthesis equals 5 x plus 3 y f left parenthesis 6 comma 0 right parenthesis equals 5 left parenthesis 6 right parenthesis plus 3 left parenthesis 0 right parenthesis f left parenthesis 6 comma 0 right parenthesis equals 30 end style  

begin mathsize 14px style rightwards double arrow left parenthesis 0 comma 6 right parenthesis end style 

begin mathsize 14px style f left parenthesis x comma y right parenthesis equals 5 x plus 3 y f left parenthesis 0 comma 6 right parenthesis equals 5 left parenthesis 0 right parenthesis plus 3 left parenthesis 6 right parenthesis f left parenthesis 0 comma 6 right parenthesis equals 18 end style 

begin mathsize 14px style rightwards double arrow left parenthesis 15 over 2 comma 0 right parenthesis end style 

begin mathsize 14px style f left parenthesis x comma y right parenthesis equals 5 x plus 3 y f open parentheses 15 over 2 comma 0 close parentheses equals 5 open parentheses 15 over 2 close parentheses plus 3 left parenthesis 0 right parenthesis f left parenthesis 0 comma 6 right parenthesis equals 75 over 2 end style 

begin mathsize 14px style rightwards double arrow left parenthesis 0 comma 5 right parenthesis end style 

begin mathsize 14px style f left parenthesis x comma y right parenthesis equals 5 x plus 3 y f left parenthesis 0 comma 5 right parenthesis equals 5 left parenthesis 0 right parenthesis plus 3 left parenthesis 5 right parenthesis f left parenthesis 0 comma 5 right parenthesis equals 15 end style 

begin mathsize 14px style rightwards double arrow left parenthesis 2 comma 3 right parenthesis end style 

begin mathsize 14px style f left parenthesis x comma y right parenthesis equals 5 x plus 3 y f left parenthesis 2 comma 3 right parenthesis equals 5 left parenthesis 2 right parenthesis plus 3 left parenthesis 3 right parenthesis f left parenthesis 2 comma 3 right parenthesis equals 19 end style 

Jadi, diperoleh bahwa nilai maksimum persamaan undefined adalah begin mathsize 14px style 75 over 2 end style 

 

 

 

 

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