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2. Verificati dacă pe sisteme de ecuatii, ap a) \( \left\{\begin{array}{l}x-\sqrt{2} y=0 \\ \sqrt{2} x+y-1=0\end{array}\right. \) b)

Ask by Donnelly Owen. in Romania
Mar 15,2025

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The solution to the system of equations is \( x = \frac{\sqrt{2}}{3} \) and \( y = \frac{1}{3} \).

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Să investigăm sistemul de ecuații propus! Poate fi util să-l scriem într-o formă mai familiară pentru a putea determina soluția. Din prima ecuație \( x = \sqrt{2} y \). Apoi, substituind această expresie în a doua ecuație, obținem \( \sqrt{2}(\sqrt{2} y) + y - 1 = 0 \), adică \( 2y + y - 1 = 0 \), ceea ce ne dă \( 3y = 1 \) sau \( y = \frac{1}{3} \). Folosind această valoare în prima ecuație, aflăm că \( x = \sqrt{2}\cdot\frac{1}{3} = \frac{\sqrt{2}}{3} \). Astfel, soluția sistemului este \( \left( \frac{\sqrt{2}}{3}, \frac{1}{3} \right) \). În cadrul acestui tip de sistem, este important să fim atenți la manipularea radicalilor. O greșeală frecventă este neglijarea semnelor negative în ecuații sau confundarea coeficientului variabilelor. Asigură-te că urmezi fiecare pas cu atenție și că verifici soluția finală în ambele ecuații pentru a confirma că totul este corect.

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