1. Preparación del sistema:
Reescribimos las ecuaciones agrupando las expresiones $(x+y)$ y $(x-y)$:
$$ \begin{cases} 12(x+y)^2 + (x+y) = 2.5 \\ 6(x-y)^2 + (x-y) = 0.125 \end{cases} $$
Convertimos los decimales a fracciones para facilitar el cálculo: $2.5 = \frac{5}{2}$ y $0.125 = \frac{1}{8}$.

2. Resolución de las ecuaciones auxiliares:
Sea $u = x+y$ y $v = x-y$.
Para $u$:
$$ 12u^2 + u - \frac{5}{2} = 0 \implies 24u^2 + 2u - 5 = 0 $$
Resolviendo: $u = \frac{-2 \pm \sqrt{4 - 4(24)(-5)}}{48} = \frac{-2 \pm 22}{48} \implies u_1 = \frac{5}{12}, u_2 = -\frac{1}{2}$.

Para $v$:
$$ 6v^2 + v - \frac{1}{8} = 0 \implies 48v^2 + 8v - 1 = 0 $$
Resolviendo: $v = \frac{-8 \pm \sqrt{64 - 4(48)(-1)}}{96} = \frac{-8 \pm 16}{96} \implies v_1 = \frac{1}{12}, v_2 = -\frac{1}{4}$.

3. Cálculo de $(x, y)$:
Usando las relaciones $x = \frac{u+v}{2}$ e $y = \frac{u-v}{2}$ para las 4 combinaciones:

• $u=\frac{5}{12}, v=\frac{1}{12} \implies x=\frac{6/12}{2} = \frac{1}{4}, y=\frac{4/12}{2} = \frac{1}{6}$
• $u=\frac{5}{12}, v=-\frac{1}{4} \implies x=\frac{2/12}{2} = \frac{1}{12}, y=\frac{8/12}{2} = \frac{1}{3}$
• $u=-\frac{1}{2}, v=\frac{1}{12} \implies x=\frac{-5/12}{2} = -\frac{5}{24}, y=\frac{-7/12}{2} = -\frac{7}{24}$
• $u=-\frac{1}{2}, v=-\frac{1}{4} \implies x=\frac{-3/4}{2} = -\frac{3}{8}, y=\frac{-1/4}{2} = -\frac{1}{8}$

Resultado Final:
$$ \boxed{(x, y) \in \left\{ \left(\frac{1}{4}, \frac{1}{6}\right), \left(\frac{1}{12}, \frac{1}{3}\right), \left(-\frac{5}{24}, -\frac{7}{24}\right), \left(-\frac{3}{8}, -\frac{1}{8}\right) \right\}} $$