How to solve a system of 3 equations for 3 unknowns

[christurnadge]

Homework Statement

The variables of interest are :

Constants are a, b, c, and g

Unknowns are x, m, and L

The intent is to define x, m, and L in terms of a, b, c, and g.

Homework Equations

The three dependent equations are :

(1) x + g*m = c

(2) [x / (a^2)] + L = 0

(3) [m / (b^2)] + g*L = 0

The Attempt at a Solution

The solution is known to be :

(4) x = [(a^2)*c] / [(a^2) + (g^2)*(b^2)]

(5) m = [(b^2)*g*c] / [(a^2) + (g^2)*(b^2)]

(6) L = (-c) / [(a^2) + (g^2)*(b^2)]

I would like to know the steps used to calculate this solution, using substitution and/or elimination. I apologise for not using LaTex formatting for the equations. Thank you in advance.

 

[Ivan92]

I can help but where is your work? Have you even tried to work this on your own?

 

[christurnadge]

I'm interested as to how this is solved as it is part of a numerical method (adjoint-states) that I'm trying to understand. I've never studied linear algebra formally, so I don't have any experience in solving systems of equations. Through internet searching I have noted the use of substitution and elimination to solve such problems, but nonetheless I do not know how to begin.

I hope that you or other forum members may be able to demonstrate the steps required to achieve the solutions presented.

Thank you.

 

[eumyang]

I hope that you or other forum members may be able to demonstrate the steps required to achieve the solutions presented.

We are not allowed to just give the solutions -- it's against forum rules.

I'll give you a hint to start, and I ask that you show the work in using the hint:
- Solve equation (2) for x.
- Solve equation (3) for m.
- Plug in the results into equation (1).

 

[christurnadge]

Using eumyang's suggestions, I can now calculate the solutions as follows :

(1) [x / (a^2)] + L = 0, therefore

(2) x = -(a^2)*L

(3) [m / (b^2)] + g*L = 0, therefore

(4) m = -(b^2)*g*L


For the following equation,

(5) x + g*m = c

using the identities provided by equations (2) and (4) gives :

(6) -(a^2)*L + g*[-(b^2)*g*L] = c

which can be rearranged to solve for L :

(7) L = (-c) / [(a^2) + (g^2)*(b^2)]


Now this definition of L can be inserted into equations (2) and (4) to provide definitions for x and m.


Thanks very much eumyang for your suggestions to get started!