Many ways to go about it. Consider if you just add the two equations together:

2x + y = 4

2x - y = 8

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4x = 12

Then solve for x, so x = 3. Plug that in to either of your original equations to find y.

An alternate solution method is to solve one of them for y. We'll use the top one:

y = 4 - 2x.

Now, take that value and put it in for y in the second one.

2x - (4 - 2x) = 8.

Simplifying that, we get

2x - 4 + 2x = 8

so

4x - 4 = 8

so

4x = 12

so x = 3.

You know y = 4 - 2x, so y = 4 - 6 = -2.