28.
maijā
Eksāmens VĒSTURĒ 12. KLASEI
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### Teorija

Ir vienādojumu sistēmas, kurā vienā no rindiņām ir savstarpēji apgriezti lielumi.
Piemēram,
$\left\{\begin{array}{l}\sqrt{\frac{x}{y}}-\sqrt{\frac{y}{x}}=\frac{5}{6}\\ x-y=5\end{array}\right\$

Nezināmo kombinācijas ir izdevīgi apzīmēt ar palīgnezināmo:
$\sqrt{\frac{x}{y}}=a\phantom{\rule{0.147em}{0ex}}⇒\phantom{\rule{0.147em}{0ex}}\sqrt{\frac{y}{x}}=\frac{1}{a}$

Jaunos lielumus ievieto sistēmas pirmajā rindiņā un iegūst $a$ vērtības.
$\begin{array}{l}a-\frac{1}{a}=\frac{5}{6}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\frac{{a}^{\left(6a}}{1}-\frac{{1}^{\left(6}}{a}=\frac{{5}^{\left(a}}{6}\phantom{\rule{0.147em}{0ex}}\\ \\ \frac{6{a}^{2}-5a-6}{6a}=0\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}6a\ne 0\\ 6{a}^{2}-5a-6=0\\ \\ {a}_{1}=-\frac{2}{3}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}{a}_{2}=\frac{3}{2}\end{array}$
Svarīgi!
Metodes būtība ir tāda, ka, izmantojot palīgnezināmo, no vienas sarežģītas sistēmas var iegūt divas - vienkāršākas sistēmas.
$\begin{array}{l}\underset{¯}{\phantom{\rule{1.176em}{0ex}}\left\{\begin{array}{l}\sqrt{\frac{x}{y}}-\sqrt{\frac{y}{x}}=\frac{5}{6}\\ x-y=5\end{array}\right\\phantom{\rule{2.205em}{0ex}}}\\ \phantom{\rule{1.911em}{0ex}}↙\phantom{\rule{2.205em}{0ex}}↘\\ \left\{\begin{array}{l}\sqrt{\frac{x}{y}}=-\frac{8}{12}\\ \phantom{\rule{0.294em}{0ex}}x-y=5\phantom{\rule{1.176em}{0ex}}\end{array}\right\\phantom{\rule{1.764em}{0ex}}\left\{\begin{array}{l}\sqrt{\frac{x}{y}}=\frac{3}{2}\\ x-y=5\end{array}\right\\end{array}$

Pirmajai sistēmai nav atrisinājuma, jo saknes vērtība nevar būt negatīvs skaitlis. Risina otro sistēmu, pirmās rindiņas abas puses kāpinot kvadrātā, izsakot $x$ un ievietojot otrajā rindiņā:
$\left\{\begin{array}{l}\sqrt{\frac{x}{y}}=\frac{3}{2}\\ x-y=5\end{array}\right\\phantom{\rule{1.323em}{0ex}}\left\{\begin{array}{l}\frac{x}{y}\overline{)=}\frac{9}{4}\\ x-y=5\end{array}\right\\phantom{\rule{1.176em}{0ex}}\left\{\begin{array}{l}x=\frac{9y}{4}\\ \frac{9y}{4}-y=5\end{array}\right\\phantom{\rule{0.294em}{0ex}}...\phantom{\rule{0.441em}{0ex}}\left\{\begin{array}{l}x=9\\ y=4\end{array}\right\$

Piemērs:
Pamēģini patstāvīgi atrisināt sistēmu ar substitūcijas metodi:
$\left\{\begin{array}{l}\frac{x}{y}+\frac{y}{x}=\frac{26}{5}\\ {x}^{2}-{y}^{2}=24\end{array}\right\\phantom{\rule{0.147em}{0ex}}⇒\phantom{\rule{0.147em}{0ex}}...\phantom{\rule{0.147em}{0ex}}⇒\left\{\begin{array}{l}{x}_{1}=5\\ {y}_{1}=1\end{array}\right\\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.588em}{0ex}}\left\{\begin{array}{l}{x}_{2}=-5\\ {y}_{2}=-1\end{array}\right\$