I) Metoda substituției :
[tex]\left\{\begin{matrix}
4x - 3y = 20\\
5x + 2y = 2\end{matrix}\right.[/tex]
[tex]4x - 3y = 20[/tex]
[tex]4x = 20 + 3y[/tex]
[tex]x = \frac{20 + 3y}{4} [/tex]
[tex]5x + 2y = 2[/tex]
[tex]5 \times \frac{20 + 3y}{4} + 2y = 2[/tex]
[tex] \frac{5(20 + 3y)}{4} + 2y = 2 \: | \times 4[/tex]
[tex]5(20 + 3y) + 8y = 8[/tex]
[tex]100 + 15y + 8y = 8[/tex]
[tex]15y + 8y = 8 - 100[/tex]
[tex]23y = - 92[/tex]
[tex]y = - \frac{92}{23} [/tex]
[tex]y = - 4[/tex]
[tex]x = \frac{20 + 3y}{4} [/tex]
[tex]x = \frac{20 + 3 \times ( - 4)}{4} [/tex]
[tex]x = \frac{20 - 12}{4} [/tex]
[tex]x = \frac{8}{4} [/tex]
[tex]x = 2[/tex]
[tex]S=\left \{ (x,y)\right \}[/tex]
[tex]S=\left \{ (2,-4)\right \}[/tex]
II) Metoda reducerii :
[tex]\left\{\begin{matrix}
4x - 3y = 20 \: | \times 2\\
5x + 2y = 2 \: | \times 3\end{matrix}\right.[/tex]
[tex]\left\{\begin{matrix}
8x - 6y = 40\\
15x + 6y = 6\end{matrix}\right.[/tex]
[tex]8x + 15x - 6y + 6y = 40 + 6[/tex]
[tex]23x = 46[/tex]
[tex]x = \frac{46}{23} [/tex]
[tex]x = 2[/tex]
[tex]4x - 3y = 20[/tex]
[tex]4 \times 2 - 3y = 20[/tex]
[tex]8 - 3y = 20[/tex]
[tex] - 3y = 20 - 8[/tex]
[tex] - 3y = 12 \: | \div ( - 3)[/tex]
[tex]y = - 4[/tex]
[tex]S=\left \{ (x,y)\right \}[/tex]
[tex]S=\left \{ (2,-4)\right \}[/tex]
