Bridge circuit - 5 eqns with 5 unknowns

[skierboy]

Homework Statement

The bridge circuit for this problem:
http://img52.imageshack.us/img52/7366/bridgecircuit.jpg

I have to derive a formula for the equivalent resistance (R_eq) of the bridge circuit shown in the link.

Homework Equations

I've been able to come up with the 5 equations with 5 unknowns using Kirchoff's loop and junction laws. They are as follows:

1) I = I₁ + I₂
2) I₁ = I₃ + I₅
3) I₄ = I₂ + I₅
4) I₁R₁ + I₅R₅ = I₂R₂
5) I₅R₅ + I₄R₄ = I₃R₃

Also, IR_eq = I₁R₁ + I₃R₃

so, R_eq = (I₁R₁ + I₃R₃)/I

The Attempt at a Solution

I am having trouble starting to solve this system of equations. I've tried various substitutions but they have all led me to dead ends.

Based on equation 6, I know that I have to express all the currents across each resistor in terms of 'I', so that 'I' cancels when doing the final substitution in equation 6, leaving only the resistors.

I am not asking for a full derivation, but simply some direction as to how to approach solving this system.

Thank you in advance!

 

[Delphi51]

You also have an equation for the current coming out: 7) I3 + I4 = I.
I would suggest using the relatively simple eqn 3 to eliminate all the I4s. That is, replace all I4 entries with I2 + I5. Likewise, solve eqn 2 for I5 and eliminate all the I5 entries. I'm trying to keep I1 and I3 so you can solve for them and use them in eqn 6. Solve eqn 1 for I2 eliminate that. The next steps, solving for I1 and I3 will be rather messy!

 

[kerimek]

I would first start by simplifying the equations as much as possible. For example, equations 2 and 3 can be combined to eliminate I2 and I3. Additionally, equations 4 and 5 can be combined to eliminate I5.

Once the equations are simplified, I would try to isolate one variable in each equation. For example, in equation 1, I can isolate I2 by subtracting I1 from both sides. Then, in equation 4, I can isolate I4 by subtracting I2 from both sides and substituting in the expression for I2 from equation 1. This will give me an equation with only one unknown, I5.

I would continue this process of isolating one variable at a time until I have equations with only one unknown each. Then, I can solve for each unknown using substitution or elimination.

Once I have the values for all the unknowns, I can use equation 6 to calculate the equivalent resistance. It may also be helpful to draw a circuit diagram and label the currents and resistors to visualize the problem better.

Overall, solving a system of equations with multiple unknowns can be challenging, but breaking it down into smaller, simpler steps can make it more manageable. It may also be helpful to consult with a colleague or reference material for additional guidance.