I can't solve this set of equations, please help me.
$$(1+i)z_1 + (1-i)z_2 = 1+i$$ $$(1-i)z_1 + (1+i)z_2 = 1+3i$$
One way is to multiply the top equation through by $(1-i)$ and the bottom one by $(1+i)$ to give
$$2z_1 - 2iz_2 = 2\qquad\qquad$$ $$2z_1 + 2iz_2 = -2+4i$$
You can now eliminate one of the unknowns and find the other. You can then substitute this back and get the complete solution.