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In interactive use, the last computed result is always displayed.

In non-interactive use, it is not. To get it displayed, use print.

var('A1, A2, N1, N2, q')
eq1 = A1 == N1/(N2*q + N1)^2
eq2 = A2 == q^2*N2/(N2*q + N1)^2
eqns = [eq1, eq2, A1 == 0.160, A2 == 0.015, N2 == 10]
solns = solve (eqns, A1, A2, q, N1, N2, solution_dict=True)
print([[s[N1].n(30), s[q].n(30)] for s in solns])