$6)$ $\left\{\begin{aligned} &3x + y = 1 \\ &x - 2y = 5 \end{aligned}\right.$

$\left\{\begin{aligned} &y = 1 - 3x \\ &x - 2(1 - 3x) = 5 \end{aligned}\right.$

$\left\{\begin{aligned} &y = 1 - 3x \\ &x - 2 + 6x = 5 \end{aligned}\right.$

$\left\{\begin{aligned} &y = 1 - 3x \\ &7x = 7 \end{aligned}\right.$

$\left\{\begin{aligned} &x = 1 \\ &y = 1 - 3 \cdot 1 \end{aligned}\right.$

$\left\{\begin{aligned} &x = 1 \\ &y = -2 \end{aligned}\right.$

Vậy hệ phương trình có nghiệm là $(x; y) = (1; -2)$

 

$7)$ $\left\{\begin{aligned} &x + 2y = 4 \\ &3x + y = 7 \end{aligned}\right.$

$\left\{\begin{aligned} &x = 4 - 2y \\ &3(4 - 2y) + y = 7 \end{aligned}\right.$

$\left\{\begin{aligned} &x = 4 - 2y \\ &12 - 6y + y = 7 \end{aligned}\right.$

$\left\{\begin{aligned} &x = 4 - 2y \\ &-5y = -5 \end{aligned}\right.$

$\left\{\begin{aligned} &y = 1 \\ &x = 4 - 2 \cdot 1 \end{aligned}\right.$

$\left\{\begin{aligned} &x = 2 \\ &y = 1 \end{aligned}\right.$

Vậy hệ phương trình có nghiệm là $(x; y) = (2; 1)$

 

$8)$ $\left\{\begin{aligned} &x + 3y = -2 \\ &5x - 4y = 3 \end{aligned}\right.$

$\left\{\begin{aligned} &x = -2 - 3y \\ &5(-2 - 3y) - 4y = 3 \end{aligned}\right.$

$\left\{\begin{aligned} &x = -2 - 3y \\ &-10 - 15y - 4y = 3 \end{aligned}\right.$

$\left\{\begin{aligned} &x = -2 - 3y \\ &-19y = 13 \end{aligned}\right.$

$\left\{\begin{aligned} &y = -\frac{13}{19} \\ &x = -2 - 3 \cdot \left(-\frac{13}{19}\right) \end{aligned}\right.$

$\left\{\begin{aligned} &x = \frac{1}{19} \\ &y = -\frac{13}{19} \end{aligned}\right.$

Vậy hệ phương trình có nghiệm là $(x; y) = \left(\frac{1}{19}; -\frac{13}{19}\right)$

 

$9)$ $\left\{\begin{aligned} &2x + y = 4 \\ &5x - 4y = 3 \end{aligned}\right.$

$\left\{\begin{aligned} &y = 4 - 2x \\ &5x - 4(4 - 2x) = 3 \end{aligned}\right.$

$\left\{\begin{aligned} &y = 4 - 2x \\ &5x - 16 + 8x = 3 \end{aligned}\right.$

$\left\{\begin{aligned} &y = 4 - 2x \\ &13x = 19 \end{aligned}\right.$

$\left\{\begin{aligned} &x = \frac{19}{13} \\ &y = 4 - 2 \cdot \frac{19}{13} \end{aligned}\right.$

$\left\{\begin{aligned} &x = \frac{19}{13} \\ &y = \frac{14}{13} \end{aligned}\right.$

Vậy hệ phương trình có nghiệm là $(x; y) = \left(\frac{19}{13}; \frac{14}{13}\right)$

 

$10)$ $\left\{\begin{aligned} &x + y = 5 \\ &2x - y = 1 \end{aligned}\right.$

$\left\{\begin{aligned} &y = 5 - x \\ &2x - (5 - x) = 1 \end{aligned}\right.$

$\left\{\begin{aligned} &y = 5 - x \\ &3x - 5 = 1 \end{aligned}\right.$

$\left\{\begin{aligned} &3x = 6 \\ &y = 5 - x \end{aligned}\right.$

$\left\{\begin{aligned} &x = 2 \\ &y = 5 - 2 \end{aligned}\right.$

$\left\{\begin{aligned} &x = 2 \\ &y = 3 \end{aligned}\right.$

Vậy hệ phương trình có nghiệm là $(x; y) = (2; 3)$

 

$11)$ $\left\{\begin{aligned} &x + 2y = 3 \\ &4x + 5y = 6 \end{aligned}\right.$

$\left\{\begin{aligned} &x = 3 - 2y \\ &4(3 - 2y) + 5y = 6 \end{aligned}\right.$

$\left\{\begin{aligned} &x = 3 - 2y \\ &12 - 8y + 5y = 6 \end{aligned}\right.$

$\left\{\begin{aligned} &x = 3 - 2y \\ &-3y = -6 \end{aligned}\right.$

$\left\{\begin{aligned} &y = 2 \\ &x = 3 - 2 \cdot 2 \end{aligned}\right.$

$\left\{\begin{aligned} &x = -1 \\ &y = 2 \end{aligned}\right.$

Vậy hệ phương trình có nghiệm là $(x; y) = (-1; 2)$