302/MR00. zadatak / Matematika - Zbirka rešenih zadataka

MR-302 / 302779. zadatak

Rešite sledeći sistem jednačina:

$\begin{matrix} x^{2} + y^{2} = 29 \\ x + y = 7 \end{matrix}$

x^{2} + y^{2} = 29 x + y = 7

y = 7 - x

x^{2} + {(7 - x)}^{2} = 29

x^{2} + 49 - 14 x + x^{2} = 29

x^{2} + 49 - 14 x + x^{2} - 29 = 0

2 x^{2} - 14 x + 20 = 0 | \frac{1}{2}

x^{2} - 7 x + 10 = 0; a = 1, b = - 7, c = 10

x_{1, 2} = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a} = \frac{7 \pm \sqrt{7^{2} - 4 \cdot 1 \cdot 10}}{2} = \frac{7 \pm \sqrt{49 - 40}}{2} = \frac{7 \pm \sqrt{9}}{2} = \frac{7 \pm 3}{2}

x_{1} = \frac{7 + 3}{2} = \frac{10}{2}; x_{2} = \frac{7 - 3}{2} = \frac{4}{2}

x_{1} = 5; x_{2} = 2

y_{1} = 7 - x_{1} = 7 - 5 = 2

y_{2} = 7 - x_{2} = 7 - 2 = 5

$(5, 2); (2, 5)$

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