Solving equations with multiple unique solutions

Questions
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1

Hi All,

I am new to Mathematica and I apologize if this is a very basic question.

I am attempting to use the solve function to derive multiple solutions for a variable, however it seems as though I can only have the same number of equations as I do unknowns, which then leads to only one solution per unknown.

E.g.:
Solve[{m == ab, n == ac, o == b*c}, {a, b, c}]

What I am trying to solve for is this:
Solve[{m == ab, n == ac, o == ad, p == bc, q == bd, r == cd}, {a, b, c, d}]

This indicates that each unknown (a,b,c,d) will have more than one unique solution. However, this statement leads to an empty solution.

I can try to break up the over-identified (determined?) statement and eventually produce all unique solutions, however this could get complicated and tiresome with a larger model. I was wondering, is there a way to get Mathematica to produce all unique solutions in a situation like this?

2

(1) ab is not the same asa*b.

(2) The second system is overdetermined. Might instead use Reduce for it.

Reduce[{m == a*b, n == a*c, o == a*d, p == b*c, q == b*d,  r == c*d}, {a, b, c, d}]

(* (o == 0 && n == 0 && m == 0 && a == 0 && 
   r != 0 && (b == -((Sqrt[p] Sqrt[q])/Sqrt[r]) || 
     b == (Sqrt[p] Sqrt[q])/Sqrt[r]) && b != 0 && c == p/b && 
   d == q/b) || (q == 0 && p == 0 && m == 0 && 
   r != 0 && (a == -((Sqrt[n] Sqrt[o])/Sqrt[r]) || 
     a == (Sqrt[n] Sqrt[o])/Sqrt[r]) && b == 0 && a != 0 && c == n/a &&
    d == o/a) || (r == 0 && p == 0 && n == 0 && 
   q != 0 && (a == -((Sqrt[m] Sqrt[o])/Sqrt[q]) || 
     a == (Sqrt[m] Sqrt[o])/Sqrt[q]) && a != 0 && b == m/a && c == 0 &&
    d == o/a) || (r == 0 && q == 0 && o == 0 && 
   p != 0 && (a == -((Sqrt[m] Sqrt[n])/Sqrt[p]) || 
     a == (Sqrt[m] Sqrt[n])/Sqrt[p]) && a != 0 && b == m/a && 
   c == n/a && d == 0) || (r == 0 && q == 0 && p == 0 && o == 0 && 
   n == 0 && m == 0 && a == 0 && b == 0 && c == 0) || (r == 0 && 
   q == 0 && p == 0 && o == 0 && n == 0 && a != 0 && b == m/a && 
   c == 0 && d == 0) || (q == 0 && p == 0 && o == 0 && n == 0 && 
   m == 0 && a == 0 && b == 0 && c != 0 && d == r/c && 
   r != 0) || (r == 0 && p == 0 && o == 0 && n == 0 && m == 0 && 
   a == 0 && c == 0 && b != 0 && d == q/b && q != 0) || (r == 0 && 
   q == 0 && o == 0 && n == 0 && m == 0 && a == 0 && b != 0 && 
   c == p/b && d == 0 && p != 0) || (r == 0 && q == 0 && p == 0 && 
   n == 0 && m == 0 && b == 0 && c == 0 && a != 0 && d == o/a && 
   o != 0) || (r == 0 && q == 0 && p == 0 && o == 0 && m == 0 && 
   b == 0 && a != 0 && c == n/a && d == 0 && n != 0) || (r == 0 && 
   q == 0 && p == 0 && o == 0 && n == 0 && m == 0 && a == 0 && 
   b == 0 && c != 0 && d == 0) || (r == 0 && q == 0 && p == 0 && 
   o == 0 && n == 0 && m == 0 && a == 0 && b != 0 && c == 0 && 
   d == 0) || (q != 0 && n == (o p)/q && r != 0 && m == (o p)/r && 
   p != 0 && (a == -((Sqrt[m] Sqrt[n])/Sqrt[p]) || 
     a == (Sqrt[m] Sqrt[n])/Sqrt[p]) && a != 0 && b == m/a && 
   c == n/a && d == o/a) *)
3

Thank you, Daniel.
I think this will work for my purposes! Looks like I can just replace the first unknown in the second phrase to get the multiple solutions for it.