Symmetric looking equations needing a symmetric solution

[FrankDrebon]

Hi all,

I have a set of equations that look very nice and symmetric, but the only way I'm able to find solutions to them is with pages and pages of algebra! Can any members with more of a mathematical flair than myself point me in the direction of a more direct and satisfactory method of solution?

The equations are:

$\frac{{A_1 }}{{A_1 + A_2 + A_3 }} = \frac{{b_1 c_1 }}{{b_1 c_1 + b_2 c_2 + b_3 c_3 }}$

$\frac{{A_2 }}{{A_1 + A_2 + A_3 }} = \frac{{b_2 c_2 }}{{b_1 c_1 + b_2 c_2 + b_3 c_3 }}$

$\frac{{A_3 }}{{A_1 + A_2 + A_3 }} = \frac{{b_3 c_3 }}{{b_1 c_1 + b_2 c_2 + b_3 c_3 }}$

Given that:

$b_1 + b_2 + b_3 = 1$

I am looking to express $b_1$, $b_2$ and $b_3$ in terms of $A_1$, $A_2$, $A_3$, $c_1$, $c_2$ and $c_3$.

In general, I guess the problem would be:

$A_i \sum\limits_j {b_j c_j } = b_i c_i \sum\limits_j {A_j }$

$\sum\limits_j {b_j } = 1$

I have solved the problem for the $j = 2$ case, with the neat looking solutions:

$b_1 = \frac{{A_1 c_2 }}{{A_1 c_2 + A_2 c_1 }}$

$b_2 = \frac{{A_2 c_1 }}{{A_1 c_2 + A_2 c_1 }}$

I'm looking for similar solutions to $j = 3$, or more...if a general solution is obvious to someone else!

Any ideas?

Thanks in advance,

FD

 

[Stephen Tashi]

I'll write your equations as:

[Eqs. 1] $$\frac{A_i}{\sum_j Aj} = \frac {b_i c_i}{\sum_j b_j c_j}$$


Introduce another variable $\theta$ defined by

[Eq. 2] $$\theta = \frac {\sum_j b_j c_j} {\sum_j A_j}$$

Multiplying the left sides of [Eqs. 1] by $\frac{\theta}{\theta}$ gives:

[Eqs. 3] $$\frac{\theta A_i}{\sum_j b_j c_j} = \frac {b_i c_i}{\sum_j b_j c_j}$$

so

[Eqs 4] $$b_i = \theta \frac{A_i}{c_i}$$


Since $\sum_j b_j = 1$ we have:

[Eq 5] $$\sum_j \theta \frac{A_j}{c_j} = \theta \sum_j {A_j}{c_j} = 1$$

Sovling for $\theta$ gives:

[Eq 6] $$\theta = \frac{1}{\sum_j \frac{A_j}{c_j}}$$

Substituting in [Eqs 4] gives

[Eqs 7] $$b_i = \frac{1}{\sum_j \frac{A_j}{c_j}} \frac{A_i}{c_i}$$

Of course, you must check none of the steps involve division by zero. Also my reasoning seems suspiciously circular!

 

[FrankDrebon]

Ahhhh, that works beautifully - nice one!

 

[asmani]

That was an elegant solution, I would like to add another one.
Dividing the sides of the i^th equation by the sides of the j^th equation gives:
$$\frac{A_i}{A_j}=\frac{b_ic_i}{b_jc_j}$$
And then:
$$b_j=\frac{A_j}{c_j}\frac{c_i}{A_i}b_i$$
Thus:
$$\sum_{j}b_j=\frac{c_i}{A_i}b_i\sum_{j}\frac{A_j}{c_j}$$
And since $\sum_{j}b_j=1$ we get:
$$b_i=\frac{A_i}{c_i}\frac{1}{\sum_{j}\frac{A_j}{c_j}}$$

 

[CellCoree]

I understand your frustration with the complexity of finding solutions to symmetric equations. Symmetry often provides a sense of simplicity and elegance, but in reality, it can make the problem more challenging to solve. However, there are several approaches that can help in finding a more direct and satisfactory solution to your equations.

Firstly, it may be helpful to consider using a computer program or mathematical software to assist in solving these equations. These tools can handle complex algebraic manipulations and can often find solutions more efficiently than manual methods. Additionally, they can provide visual representations of the solutions, which can aid in understanding the patterns and relationships within the equations.

Another approach is to use symmetry itself to your advantage. Often, symmetric equations can be simplified by taking advantage of the symmetry and reducing the number of variables or equations to be solved. In your case, the given condition b_1 + b_2 + b_3 = 1 can be used to eliminate one of the variables, reducing the problem to solving for only two variables instead of three.

Furthermore, it may be beneficial to explore different mathematical techniques such as substitution or elimination to simplify the equations and reduce the number of steps needed to find a solution. It may also be helpful to break down the problem into smaller, more manageable parts and solve them separately before combining the solutions to find the overall solution.

In summary, finding a direct and satisfactory solution to symmetric equations can be challenging, but there are various approaches that can assist in simplifying the problem and finding solutions more efficiently. I hope these suggestions are helpful, and I wish you success in finding a solution to your equations.