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You can add a new variable to model D/I:

sage: var('D,P,I,z')
sage: e = (D*P - D+ I)/(D*(P-1))
sage: e
(D*p - D + I)/(D*(P - 1))
sage: solve([e==0,z==D/I],z,D)
[[z == -1/(P - 1), D == -I/(P - 1)]]

You can add a new variable to model D/I and solve a system of equations :

sage: var('D,P,I,z')
sage: e = (D*P - D+ I)/(D*(P-1))
D + I)/(D*(P - 1))
sage: e
(D*p - D + I)/(D*(P - 1))
sage: solve([e==0,z==D/I],z,D)
solve([e == 0, z == D/I], z, D)
[[z == -1/(P - 1), D == -I/(P - 1)]]