Question

Aflați soluția sistemului
{4x-3y=20.
{5x+2y=2

Answer 1

I) Metoda substituției :

$\left\{\begin{matrix} 4x - 3y = 20\\ 5x + 2y = 2\end{matrix}\right.$

$4x - 3y = 20$

$4x = 20 + 3y$

$x = \frac{20 + 3y}{4}$

$5x + 2y = 2$

$5 \times \frac{20 + 3y}{4} + 2y = 2$

$\frac{5(20 + 3y)}{4} + 2y = 2 \: | \times 4$

$5(20 + 3y) + 8y = 8$

$100 + 15y + 8y = 8$

$15y + 8y = 8 - 100$

$23y = - 92$

$y = - \frac{92}{23}$

$y = - 4$

$x = \frac{20 + 3y}{4}$

$x = \frac{20 + 3 \times ( - 4)}{4}$

$x = \frac{20 - 12}{4}$

$x = \frac{8}{4}$

$x = 2$

$S=\left \{ (x,y)\right \}$

$S=\left \{ (2,-4)\right \}$

II) Metoda reducerii :

$\left\{\begin{matrix} 4x - 3y = 20 \: | \times 2\\ 5x + 2y = 2 \: | \times 3\end{matrix}\right.$

$\left\{\begin{matrix} 8x - 6y = 40\\ 15x + 6y = 6\end{matrix}\right.$

$8x + 15x - 6y + 6y = 40 + 6$

$23x = 46$

$x = \frac{46}{23}$

$x = 2$

$4x - 3y = 20$

$4 \times 2 - 3y = 20$

$8 - 3y = 20$

$- 3y = 20 - 8$

$- 3y = 12 \: | \div ( - 3)$

$y = - 4$

$S=\left \{ (x,y)\right \}$

$S=\left \{ (2,-4)\right \}$