Bank Soal Matematika SMA Sistem Persamaan Dua Variabel

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Himpunan penyelesaian $\left(x,y\right)$ untuk sistem persamaan kuadrat-kuadrat dua variabel di atas adalah ....

A

$HP=\left\{\left(\sqrt{\frac{8}{19}},\sqrt{8}\right),\left(-\sqrt{\frac{8}{19}},\sqrt{8}\right),\left(\sqrt{\frac{8}{19}},-\sqrt{8}\right)\left(-\sqrt{\frac{8}{19}},-\sqrt{8}\right)\right\}$

$HP=\left\{\left(\sqrt{\frac{8}{18}},\sqrt{\frac{7}{19}}\right),\left(-\sqrt{\frac{8}{18}},\sqrt{\frac{7}{19}}\right),\left(\sqrt{\frac{38}{22}},-\sqrt{\frac{7}{19}}\right),\left(-\sqrt{\frac{38}{22}},-\sqrt{\frac{7}{19}}\right)\right\}$

C

$HP=\left\{\left(\sqrt{\frac{1}{9}},\sqrt{\frac{7}{19}}\right),\left(-\sqrt{\frac{1}{9}},\sqrt{\frac{7}{19}}\right),\left(\sqrt{\frac{1}{9}},-\sqrt{\frac{7}{19}}\right),\left(-\sqrt{\frac{1}{9}},-\sqrt{\frac{7}{19}}\right)\right\}$

D

$HP=\left\{\left(\sqrt{\frac{6}{8}},\sqrt{\frac{7}{9}}\right),\left(-\sqrt{\frac{6}{8}},\sqrt{\frac{7}{9}}\right),\left(\sqrt{\frac{6}{8}},-\sqrt{\frac{7}{9}}\right),\left(-\sqrt{\frac{6}{8}},-\sqrt{\frac{7}{9}}\right)\right\}$

E

$HP=\left\{\left(\sqrt{\frac{3}{22}},\sqrt{\frac{6}{19}}\right),\left(-\sqrt{\frac{3}{22}},\sqrt{\frac{6}{19}}\right),\left(\sqrt{\frac{3}{22}},-\sqrt{\frac{6}{19}}\right),\left(-\sqrt{\frac{3}{22}},-\sqrt{\frac{6}{19}}\right)\right\}$

Pembahasan:

Langkah-langkah dalam menyelesaikan persoalan sistem persamaan kuadrat-kuadrat dua variabel dalam bentuk pecahan adalah

Memisalkan bentuk pecahan sehingga memperoleh sistem persamaan linear dua variabel

Misalkan $\frac{1}{x^2}=a$ dan $\frac{1}{y^2}=b$ sehingga diperoleh sistem persamaan

Melakukan eliminasi pada dua persamaan

Substitusi nilai-nilai $b$ ke salah satu persamaan

$a-3b=2$

$a-3\left(\frac{1}{8}\right)=2$

$a-\frac{3}{8}=2$

$a=2+\frac{3}{8}$

$a=\frac{16+3}{8}$

$a=\frac{19}{8}$

Mencari nilai $x$ dan $y$ dari permisalan awal

$\frac{1}{x^2}=a$

$\frac{1}{x^2}=\frac{19}{8}$

$x^2=\frac{8}{19}$

$x=\pm\sqrt{\frac{8}{19}}$

$\frac{1}{y^2}=b$

$\frac{1}{y^2}=\frac{1}{8}$

$y^2=8$

$y=\pm\sqrt{8}$

Sehingga diperoleh solusi $\left(\sqrt{\frac{8}{19}},\sqrt{8}\right),\left(-\sqrt{\frac{8}{19}},\sqrt{8}\right),\left(\sqrt{\frac{8}{19}},-\sqrt{8}\right)\left(-\sqrt{\frac{8}{19}},-\sqrt{8}\right)$ atau $HP=\left\{\left(\sqrt{\frac{8}{19}},\sqrt{8}\right),\left(-\sqrt{\frac{8}{19}},\sqrt{8}\right),\left(\sqrt{\frac{8}{19}},-\sqrt{8}\right)\left(-\sqrt{\frac{8}{19}},-\sqrt{8}\right)\right\}$