What kind of problem is this and what tools can we use to tackle it?

[LuculentCabal]

Basically I have the following system:

$$ac = A$$

$$ad + bc = B$$

$$bd = C$$


where $$A$$, $$B$$, and $$C$$ are constants.

Solving for $$a, b, c,$$ and $$d$$, what kind of problem/system am I encountering and what appropriate tools (vectors and/or numerical methods perhaps[?]) would help to find the set of solutions for $$a, b, c,$$ and $$d$$?

I know that the system factors a trinomial but I don't know what kind of problem is presented by the system itself.

Thanks in advance for any insight.

 

[betel]

Inserting the first and third equation into the second gives you a quadratic equation (altghough for a product of two unknowns).
You will still end up with one unknown variable as you do not have enough equations.

 

[LuculentCabal]

You will still end up with one unknown variable as you do not have enough equations.

Could there still be numerical methods for finding the solutions?

Thanks for your reply.

 

[mikeph]

My messing around on MATLAB. Not sure how useful it is...

EDU>> syms a b c A B C
EDU>> sol = solve('a*c = A','a*d + b*c = B','b*d = C');
EDU>> [a;sol.b(1);sol.c(1);sol.d(1)]
ans =
                                 a
 a*(1/2*B-1/2*(B^2-4*A*C)^(1/2))/A
                               A/a
   (1/2*B+1/2*(B^2-4*A*C)^(1/2))/a
EDU>> [a;sol.b(2);sol.c(2);sol.d(2)]
ans =
                                 a
 a*(1/2*B+1/2*(B^2-4*A*C)^(1/2))/A
                               A/a
   (1/2*B-1/2*(B^2-4*A*C)^(1/2))/a

I guess that's two sets of solutions?

edit - I generally prefer linear algebra to sets of equations where variables are multiplied. It's kind of a nightmare actually - I can see a uniqueness/existence problem if a = 0 right away, or B^2-4*A*C < 0? What a mess...

 

[Gerenuk]

I recommend substituting the top and bottom into the middle.
$$Cx+\frac{A}{x}=B$$
(where $x=\frac{a}{b}$) and solve for x.

The variable b is arbitrary (but not equal 0) and $a=xb$
Use top and bottom equation to find rest.

EDIT: just as bebel said :D

 

[LuculentCabal]

Thanks for the output MikeyW, your MATLAB solution looks consistent with what I have found. I have found that the system can be simplified to a function of $$b, c$$, and the constants. Letting one of the variables assume any value, I can solve for the other with the quadratic formula or Newtons method, and then find the other two variables.

After simplifying:

$${b^2}{c^2} - Bbc + AC = 0$$

$$a =\frac{A}{c}$$

$$d = \frac{C}{b}$$

I don't know if this would necessarily work for higher degree polynomials (especially an odd number of variables), and wonder if there is a solution using vectors/linear algebra. My linear algebra knowledge is still elementary and haven't had to encounter multiple variables before so I don't know if there are tools that can help.

 

[LuculentCabal]

A little more complex would be the following system which factors a quintic when $$a, b, c, d, e, f, g, h, i$$ and $$j$$ are solved for with $$A, B, C, D, E,$$ and $$F$$ being constants.


$$acegi = A$$

$$acegj + acehi + acfgi + egadi + egbci = B$$

$$acehj + acfgj + egadj + egbcj + acfhi + adehi + adfgi + bcehi + bcfgi + bdegi = C$$

$$acfhj + adehj + bcehj + bcfgj + bdegj + adfhi + bcfhi + ehbdi + fgbdi + adfgj = D$$

$$adfhj + bcfhj + ehbdj + fgbdj + bdfhi = E$$

$$bdfhj = F$$


Is there any way to solve for the above ten variables with a method more streamlined than the "plug and chug" method used previously for the quadratic?

*I am not as interested in any solution as I am in any algorithm ("plug and chug" or other)

Thanks again for any info. It is greatly appreciated.