$\left\{\begin{array}{l}x+2 y=6 \\ 3 x-y=4\end{array}\right.$

Step 1 ：\(\left\{\begin{array}{l}x+2 y=6 \\ 3 x-y=4\end{array}\right.\)

Step 2 ：\(\begin{cases} 3(x+2y)=3(6) \\ 2(3x-y)=2(4) \end{cases}\)

Step 3 ：\(\begin{cases} 3x+6y=18 \\ 6x-2y=8 \end{cases}\)

Step 4 ：Add the two equations to eliminate y:

Step 5 ：\( (3x+6y) + (6x-2y) = 18 + 8 \)

Step 6 ：\( 9x + 4y = 26 \)

Step 7 ：\( 9x = 26 - 4y \)

Step 8 ：\( x = \frac{26 - 4y}{9} \)

Step 9 ：Substitute the value of x in the first equation:

Step 10 ：\( \frac{26 - 4y}{9} + 2y = 6 \)

Step 11 ：\( 26 - 4y + 18y = 54 \)

Step 12 ：\( 14y = 28 \)

Step 13 ：\( y = \frac{28}{14} \)

Step 14 ：\(\boxed{y = 2}\)

Step 15 ：Substitute the value of y in the equation for x:

Step 16 ：\( x = \frac{26 - 4(2)}{9} \)

Step 17 ：\( x = \frac{26 - 8}{9} \)

Step 18 ：\( x = \frac{18}{9} \)

Step 19 ：\(\boxed{x = 2}\)

Step 20 ：Final Answer: \(\boxed{x = 2, y = 2}\)