Question

Rezolvati urmatoarele sisteme de ecuatii:

Answer 1

Explicație pas cu pas:

a)

$\left \{ {{ \frac{x}{6} = \frac{y}{3} } \atop { \frac{x}{10} = \frac{y}{5} }} \right. \iff \left \{ {{ x = 2y} \atop { x = 2y }} \right.$

$\implies \left \{ {{x \in \mathbb{R}; y \in \mathbb{R}} \atop {x = 2y}} \right.$

b)

$\left \{ {{ \frac{6 + x}{5} = \frac{y + 12}{10} } \atop { \frac{y - 1}{3} = - \frac{x + 7}{6} }} \right. \iff \left \{ { {2(6 + x) = y + 12 } \atop { 2(y - 1) = -(x + 7)}} \right.$

$\left \{ { {12 + 2x = y + 12 } \atop { 2y - 2 = - x - 7}} \right.\iff \left \{ { {2x - y = 0} \atop { x + 2y = - 5}} \right.$

$\iff \left \{ { {2x - y = 0} \atop { - 2x - 4y =10}} \right.$

$- 5y = 10 \implies y = - 2$

$2x - ( - 2) = 0\iff 2x = - 2 \\ \implies x = - 1$

c)

$\left \{ {{ \frac{x}{2} + 2 = \frac{y + 5}{2}} \atop {x + \frac{y}{2} = \frac{9}{2} }} \right. \iff \left \{ {{ x + 4 = y + 5} \atop {2x + y = 9}} \right.$

$\left \{ {{ x - y = 1} \atop {2x + y = 9}} \right.$

$3x = 10 \implies x = \frac{10}{3}$

$\frac{10}{3} - y = 1 \iff y = \frac{10}{3} - 1 \\ \implies y = \frac{7}{3}$

d)

$\left \{ {{ \frac{x + 1}{2} + 2y = \frac{ - 3x + y}{4}} \atop { \frac{x - 1}{3} + \frac{y + 1}{2} = 0}} \right. \iff \left \{ {{2(x + 1) + 8y = - 3x + y} \atop {2(x - 1) + 3(y + 1) = 0}} \right.$

$\left \{ {{2x + 2 + 8y = - 3x + y} \atop {2x - 2 + 3y + 3 = 0}} \right. \iff \left \{ {{5x + 7y = - 2} \atop {2x + 3y = - 1}} \right.$

$\left \{ {{10x + 14y = - 4} \atop { - 10x - 15y = 5}} \right.$

$- y = 1 \implies y = - 1$

$2x + 3( - 1) = - 1\iff 2x - 3 = - 1 \\ \implies x = 1$