To solve the system of equations:
2014x + y = 2013
x + 2014y = -2013

We can use the method of substitution or elimination.

Using the method of substitution:
From the second equation, we can solve for x:
x = -2014y - 2013

Substitute this value of x into the first equation:
2014(-2014y - 2013) + y = 2013
-4074196y - 4074192 + y = 2013
-4074195y = 2013 + 4074192
-4074195y = 4076205
y = 4076205 / -4074195
y = -1

Substitute the value of y back into the second equation:
x + 2014(-1) = -2013
x - 2014 = -2013
x = -2013 + 2014
x = 1

So the solution to the system of equations is x = 1 and y = -1.