We have the system of equations:

x - 2y = 2

3x + y = 6

From equation 1 we express y.

x - 2y = 2

We pass the addend with the variable x from the left to the right and change the sign.

-2y = 2 - x

We divide both parts of the equation by the multiplier of y.

[tex]\dfrac{(-1)2y}{-2}=\dfrac{2-x}{-2}[/tex]

[tex]y=\dfrac{x}{2}-1[/tex]

We put the result y in equation 2.

3x + y = 6

We obtain:

[tex]3x+\left(\dfrac{x}{2}-1\right)=6[/tex]

[tex]\dfrac{7x}{2}-1=6[/tex]

We pass the free addend -1 from the left to the right, changing the sign.

[tex]\dfrac{7x}{2}=1+6[/tex]

[tex]\dfrac{7x}{2}=2[/tex]

We divide both parts of the equation by the multiplier of x.

[tex]\dfrac{\frac{7}{2}x }{\frac{7}{2} }=\dfrac{7}{\frac{7}{2}}[/tex]

[tex]x=2[/tex]

What

[tex]y=\dfrac{x}{2}-1[/tex]

then

[tex]y=-1+\dfrac{1}{2}\cdot2[/tex]

[tex]y=0[/tex]

- y = 0

- x = 2

Therefore, the correct option is C.

[tex]\red{\boxed{\green{\boxed{\boldsymbol{\purple{Pisces04}}}}}}[/tex]