IPython console for SymPy 0.7.1-git (Python 2.7.2-64-bit) (ground types: gmpy)
In [1]: var('r,d,l,k,u,a')
Out[1]: (r, d, l, k, u, a)
In [2]: system = [Eq(r*d, l*k), Eq(u*a, l*k), Eq(k, a*d)]
In [3]: system
Out[3]: [d⋅r = k⋅l, a⋅u = k⋅l, k = a⋅d]
In [4]: solve(system, a, d, l)
Out[4]:
⎡⎛ _____ _____ ⎞ ⎛ _____ _____ ⎞⎤
⎢⎜ ╱ r⋅u ╱ r⋅u ⎟ ⎜ ╱ r⋅u ╱ r⋅u ⎟⎥
⎢⎜-k⋅ ╱ ─── -k⋅ ╱ ─── _____⎟ ⎜k⋅ ╱ ─── k⋅ ╱ ─── _____⎟⎥
⎢⎜ ╲╱ k ╲╱ k ╱ r⋅u ⎟ ⎜ ╲╱ k ╲╱ k ╱ r⋅u ⎟⎥
⎢⎜────────────, ────────────, - ╱ ─── ⎟, ⎜───────────, ───────────, ╱ ─── ⎟⎥
⎣⎝ u r ╲╱ k ⎠ ⎝ u r ╲╱ k ⎠⎦
In [5]: solve(system, [a, d, l])
Out[5]:
⎡⎛ _____ _____ ⎞ ⎛ _____ _____ ⎞⎤
⎢⎜ ╱ r⋅u ╱ r⋅u ⎟ ⎜ ╱ r⋅u ╱ r⋅u ⎟⎥
⎢⎜-k⋅ ╱ ─── -k⋅ ╱ ─── _____⎟ ⎜k⋅ ╱ ─── k⋅ ╱ ─── _____⎟⎥
⎢⎜ ╲╱ k ╲╱ k ╱ r⋅u ⎟ ⎜ ╲╱ k ╲╱ k ╱ r⋅u ⎟⎥
⎢⎜────────────, ────────────, - ╱ ─── ⎟, ⎜───────────, ───────────, ╱ ─── ⎟⎥
⎣⎝ u r ╲╱ k ⎠ ⎝ u r ╲╱ k ⎠⎦
In [6]: [ eq.subs(zip((a, d, l), _5[0])).simplify() for eq in system ]
Out[6]:
⎡ _____ _____ _____ _____ ⎤
⎢ ╱ r⋅u ╱ r⋅u ╱ r⋅u ╱ r⋅u ⎥
⎢-k⋅ ╱ ─── = -k⋅ ╱ ─── , -k⋅ ╱ ─── = -k⋅ ╱ ─── , k = k⎥
⎣ ╲╱ k ╲╱ k ╲╱ k ╲╱ k ⎦
In [7]: [ bool(eq.subs(zip((a, d, l), _5[0])).simplify()) for eq in system ]
Out[7]: [True, True, True]
In [8]: [ bool(eq.subs(zip((a, d, l), _5[1])).simplify()) for eq in system ]
Out[8]: [True, True, True]
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