Tentukan semua peneyelesaian (x,y) bilangan real dari setiap sistem persamaan berikut. {x21​−y23​=14x22​+y21​=35​

Pertanyaan

Tentukan semua peneyelesaian open parentheses x comma y close parentheses bilangan real dari setiap sistem persamaan berikut.

open curly brackets table attributes columnalign left end attributes row cell 1 over x squared minus 3 over y squared equals 14 end cell row cell 2 over x squared plus 1 over y squared equals 35 end cell end table close

D. Wahyu

Master Teacher

Mahasiswa/Alumni Universitas Negeri Semarang

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Jawaban

solusi  dari sistem persamaan ini adalah  open parentheses square root of 4 over 77 end root comma space square root of 4 over 7 end root close parentheses space space dan space space open parentheses negative square root of 4 over 77 end root comma space minus square root of 4 over 7 end root close parentheses space

Pembahasan

Diketahui:

1 over x squared minus 3 over y squared equals 14 fraction numerator y squared minus 3 x squared over denominator x squared y squared end fraction equals 14 space left parenthesis Dikalikan space silang right parenthesis y squared minus 3 x squared equals 14 x squared y squared space... left parenthesis 1 right parenthesis  2 over x squared plus 1 over y squared equals 35 fraction numerator 2 y squared plus x squared over denominator x squared y squared end fraction equals 35 space left parenthesis Dikalikan space silang right parenthesis 2 y squared plus x squared equals 35 x squared y squared space... left parenthesis 2 right parenthesis

Eliminasikan persamaan (1) dan (2), sehingga

y squared minus 3 x squared equals 14 x squared y squared space space space open vertical bar x 2 close vertical bar space space space up diagonal strike 2 y squared end strike minus 3 x squared equals 28 x squared y squared space 2 y squared plus x squared equals 35 x squared y squared space space space open vertical bar x 1 close vertical bar space space space up diagonal strike 2 y squared end strike plus x squared equals 35 x squared y squared space minus 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 top enclose space space space up diagonal strike minus 4 up diagonal strike x squared end strike equals up diagonal strike minus 7 space up diagonal strike space x squared end strike y squared space space space space space space space space space space space space space space space space space space space space space space 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 space space space space space space space space space 4 space space equals 7 y squared 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 space space space space space space space space y squared equals 4 over 7 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 space space space space space space space space y equals plus-or-minus square root of 4 over 7 end root 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 space space space space space y subscript 1 equals square root of 4 over 7 space end root space comma space space space space space y subscript 2 equals negative square root of 4 over 7 space end root space space space space space space space space space space space space space

Selanjutynya, subtitusikan nilai  y subscript 1 space space dan space space y subscript 2  ke persamaan (1), sehingga

open parentheses square root of 4 over 7 end root close parentheses squared minus 3 x squared equals 14 x squared times open parentheses square root of 4 over 7 end root close parentheses squared 4 over 7 minus 3 x squared equals up diagonal strike 14 space 2 end strike x squared times fraction numerator 4 over denominator up diagonal strike 7 end fraction 4 over 7 minus 3 x squared equals 8 x squared 4 over 7 equals 8 x squared plus 3 x squared 4 over 7 equals 11 x squared x squared equals 4 over 7 colon 11 x squared equals 4 over 7 cross times 1 over 11 x squared equals 4 over 77 x equals plus-or-minus square root of 4 over 77 end root x subscript 1 equals square root of 4 over 77 end root space comma space x subscript 2 equals negative square root of 4 over 77 end root

Jadi, solusi  dari sistem persamaan ini adalah  open parentheses square root of 4 over 77 end root comma space square root of 4 over 7 end root close parentheses space space dan space space open parentheses negative square root of 4 over 77 end root comma space minus square root of 4 over 7 end root close parentheses space

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