Pertanyaan

Find the coordinate of the points of intersection of the line 5 x = 12 + 3 y and y 3 x − x 2 y = 1 .

Find the coordinate of the points of intersection of the line $5 x = 12 + 3 y$ and $\frac{3 x}{y} - \frac{2 y}{x} = 1$ .

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Klaim

D. Wahyu

Master Teacher

Mahasiswa/Alumni Universitas Negeri Semarang

Jawaban terverifikasi

Pembahasan

Diketahui: Subtitusikan persamaan (1) ke persamaan (2) yaitu Dengan menggunakan rumus ABC yaitu Dari persamaan (3) nilai variabelmasing-masing adalah sehingga, Dimana, nilai adalah Selanjutnya,menemukan nilai dengan subtitusikan nilai ke persamaan (1), sehingga Jadi, koordinat-koordinat adalah (3,42,170) dan (1,38,-1,70).

Diketahui:

$5 x equals 12 plus 3 y x equals fraction numerator 12 plus 3 y over denominator 5 end fraction... left parenthesis 1 right parenthesis fraction numerator 3 x over denominator y end fraction minus fraction numerator 2 y over denominator x end fraction equals 1 fraction numerator 3 x squared minus 2 y squared over denominator x y end fraction equals 1 space left parenthesis Dikalikan space silang right parenthesis 3 straight x squared minus 2 straight y squared equals x y... left parenthesis 2 right parenthesis$

Subtitusikan persamaan (1) ke persamaan (2) yaitu

$table attributes columnalign right center left columnspacing 0px end attributes row cell 3 open parentheses fraction numerator 12 plus 3 y over denominator 5 end fraction close parentheses squared minus 2 y squared end cell equals cell open parentheses fraction numerator 12 plus 3 y over denominator 5 end fraction close parentheses times y end cell row cell 3 open parentheses fraction numerator 144 plus 36 y plus 9 y squared over denominator 25 end fraction close parentheses minus 2 y squared end cell equals cell fraction numerator 12 y plus 3 y squared over denominator 5 end fraction end cell row cell fraction numerator 432 plus 216 y plus 27 y squared over denominator 25 end fraction minus 2 y squared end cell equals cell fraction numerator 12 y plus 3 y squared over denominator 5 end fraction end cell row cell fraction numerator 432 plus 216 y plus 27 y squared minus 50 y squared over denominator 25 end fraction end cell equals cell fraction numerator 12 y plus 3 y squared over denominator 5 end fraction space left parenthesis Kedua space ruas space dikali space 25 right parenthesis end cell row cell 432 plus 216 y minus 23 y squared end cell equals cell fraction numerator 12 y plus 3 y squared over denominator 5 end fraction space left parenthesis Kedua space ruas space dikali space 5 right parenthesis end cell row cell 5 times left parenthesis 432 plus 216 y minus 23 y squared right parenthesis end cell equals cell 12 y plus 3 y squared end cell row cell 2160 plus 1080 y minus 115 y squared end cell equals cell 12 y plus 3 y squared end cell row cell 2160 plus 1080 y minus 12 y minus 115 y squared minus 3 y squared end cell equals 0 row cell 2160 plus 1068 y minus 118 y squared end cell equals cell 0 space left parenthesis space Kedua space ruas space dibagi space 2 right parenthesis space end cell row cell 1080 plus 534 y minus 59 y squared end cell equals cell 0 space end cell row cell negative 59 y squared plus 534 y plus 1080 end cell equals cell 0 space left parenthesis Kedua space ruas space dikali space left parenthesis negative right parenthesis right parenthesis end cell row cell 59 y squared minus 534 y minus 1080 end cell equals cell 0 space... left parenthesis 3 right parenthesis end cell row blank blank blank end table$

Dengan menggunakan rumus ABC yaitu

$y subscript 1 comma 2 end subscript equals fraction numerator negative b plus-or-minus square root of b squared minus 4 times a times c end root over denominator 2 a end fraction$

Dari persamaan (3) nilai variabel masing-masing adalah

a equals 59 b equals negative 534 c equals negative 1080

sehingga,

$table attributes columnalign right center left columnspacing 0px end attributes row cell y subscript 1 comma 2 end subscript end cell equals cell fraction numerator negative left parenthesis negative 534 right parenthesis plus-or-minus square root of left parenthesis negative 534 right parenthesis squared minus 4 times left parenthesis 59 right parenthesis times left parenthesis negative 1080 right parenthesis end root over denominator 2 times 59 end fraction end cell row blank equals cell fraction numerator 534 plus-or-minus square root of 285.156 plus 254.880 end root over denominator 118 end fraction end cell row blank equals cell fraction numerator 534 plus-or-minus square root of 540.036 end root over denominator 118 end fraction end cell row blank equals cell fraction numerator 534 plus-or-minus 734 comma 871 over denominator 118 end fraction end cell end table$

Dimana, nilai y subscript 1 space dan space y subscript 2 space adalah

$table attributes columnalign right center left columnspacing 0px end attributes row cell y subscript 1 end cell equals cell fraction numerator 534 plus 734 comma 971 over denominator 118 end fraction end cell row blank equals cell 1 comma 70 end cell row cell y subscript 2 end cell equals cell fraction numerator 534 minus 734 comma 971 over denominator 118 end fraction end cell row blank equals cell negative 1 comma 70 end cell end table$

Selanjutnya,menemukan nilai x subscript 1 space dan space x subscript 2 space dengan subtitusikan nilai y subscript 1 space dan space y subscript 2 space ke persamaan (1), sehingga

$table attributes columnalign right center left columnspacing 0px end attributes row cell x subscript 1 end cell equals cell fraction numerator 12 plus 3 open parentheses 1 comma 70 close parentheses over denominator 5 end fraction end cell row blank equals cell fraction numerator 12 plus 5 comma 1 over denominator 5 end fraction end cell row blank equals cell 3 comma 42 end cell row blank blank blank row cell x subscript 1 end cell equals cell fraction numerator 12 minus 3 open parentheses 1 comma 70 close parentheses over denominator 5 end fraction end cell row blank equals cell fraction numerator 12 minus 5 comma 1 over denominator 5 end fraction end cell row blank equals cell 1 comma 38 end cell end table$

Jadi, koordinat-koordinat adalah (3,42,170) dan (1,38,-1,70).

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