equation solving - How to eliminate variables when using Solve[]

In certain problems, we need to solve systems of equations and get results in terms of just selected variables. For example, how could we solve eqn==0 below for c3 and c4 expressed in terms of c1 and c2 only, without a1 or a2?

eqn = {{c1, c2}, {c1, c3}, {c1, c4}, {c2, c3}}.{a1, a2} - {5, 2, -4, -3}

We can select two equations from the system and solve them for a1 and a2, then substitute those results back in...

asoln = Solve[eqn[[{1, 2}]] == 0, {a1, a2}];
b = eqn /. asoln;
Solve[b == 0, {c3, c4}]

(*  {{c3 -> 1/5 (3 c1 + 2 c2), c4 -> 1/5 (9 c1 - 4 c2)}} *)

This approach works but it requires that we find a subset of equations from which a1 and a2 can be solved for unambiguously, which might be difficult. Is it possible to make Solve[] eliminate a1 and a2 for us?

Answer

It turns out Solve[] has a feature that doesn't appear in the online documentation that I could find. A third argument can be added, a list of variables to be eliminated from the solution:

Solve[eqns == 0, {c3, c4}, {a1, a2}]

This yield the same output as above. And I have tested it on problems where solving for a1 and a2 (in order to eliminate them from the system) requires a careful choice of equation subset.

Reduce[] has an analogous third argument discussed here: Behavior of Reduce with variables as domain

Comments

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