Simplifying 3 simultaneous equation into one equation

[Budana]

Dear forum,
I'm new here (living in Arizona). I am stuck getting to derive this:
F1−F2−F3 = 0, (2.1)
F1c1−F2c2−F3c3 = 0, (2.2)
F1z1−F2z2−F3z3 = 0. (2.3)

Where the only unknowns are F2 and F3. The textbook states that F1, F2 and F3 can be eliminated to get:

c1z3 −c1z2+c2z1 −c2z3 −c3z1 +c2z2 = 0 (2.4)

Can anyone help the step-by-step solution in getting from 2.1 to 2.3 to get 2.4?
This is a problem of chemical process data balancing (reconciliation), but without the understanding on how to derive the mathematics, one will be stuck with future similar problems.
Thank you for your help, forum!

 

[BiGyElLoWhAt]

I haven't tried this, but assuming that your book is correct, you should be able to use this:
https://www.math.hmc.edu/calculus/tutorials/linearsystems/

 

[fresh_42]

I assume the last c2 in your post should be a c3. Anyway.
If you multiply (2.1) by $-c_1$ and add this to (2.2) you will get $(c_1-c_2)F_2 + (c_1-c_3)F_3 = 0$ and equally for the $z_i$ (multiply (2.1) by $-z_1$ and add to (2.3)).
Now we know that there must be a solution for the $F_j$ that is not all $0$.
Hence we get for our equation system
$$\begin{bmatrix} (c_1-c_2) && (c_1-c_3) \\ (z_1-z_2) && (z_1-z_3)\end{bmatrix} \cdot \begin{bmatrix} F_2 \\ F_3 \end{bmatrix} = \begin{bmatrix} 0\\0\end{bmatrix}$$
a non zero solution, which can only happen, if the determinant of our matrix is zero.
(2.4) is exactly this determinant. (Actually it's negative, but that doesn't matter because it's $0$ anyway.)

 

[Budana]

Appreciate very much the help, but still unclear honestly.
Please see page 18 of this pdf link (or page 4 of the file):
http://www.springer.com/cda/content/document/cda_downloaddocument/9781849961059-c1.pdf
The link from BiGy... solves for X1, X2, X3, whereas in this case we are eliminating F1,F2 and F3.
fresh_42 also did not reveal the solution as F2 and F3 are still in the matrix.
So again, how do we get to :
c1z3 - c1z2 + c2z1 - c2z3 - c3z1 + c2z2 = 0 ?

Thanks much again guys for your time.

 

[fresh_42]

I eliminated $F_1$ manually, because it was pretty easy and a $(2 \times 2)$ matrix is far easier than a $(3 \times 3)$ matrix.
$F_2 \, , \, F_3$ are the variables, the $x$'s if you like. They are not in the matrix.

The equation system is $A \cdot x = 0$ where $A = \begin{bmatrix} (c_1-c_2) && (c_1-c_3) \\ (z_1-z_2) && (z_1-z_3)\end{bmatrix}$ and $x = \begin{bmatrix} F_2 \\ F_3 \end{bmatrix}$.

The determinant of $A$ is $det(A) = (c_1-c_2)(z_1-z_3) - (z_1-z_2)(c_1-c_3)$.

$A$ is invertible if and only if there is only one solution, namely $x_1 = F_2 = 0$ and $x_2 = F_3 = 0$ if and only if $det(A) \neq 0$.

I now assumed there is a solution for $F_2$ and $F_3$ not both zero, which in turn would imply $F_1=0$, too, by (2.1)

(This assumption has to be reasoned by the special quantities the $F_i$ stand for. It cannot be made by math.)

So if there is such a solution, then $A$ cannot be invertible and $det(A) = 0$.
If we now multiply $0 = det(A) = (c_1-c_2)(z_1-z_3) - (z_1-z_2)(c_1-c_3)$ then we get (2.4) as required.

 

[Budana]

Impressive, fresh_42.
Could you please suggest one or two good books (or website links) on this subject so that we could refresh ourselves with more exercises?
Thanks again.

PS. and you did also find an errata in the book (equation 2.4).

 

[fresh_42]

I could recommend this:

https://www.amazon.com/dp/0387901108/?tag=pfamazon01-20

However, it's new not quite cheap but available as second hand book. That it's old doesn't matter.
If I look for e-books on the internet, I tend to find stuff with a copyright which I will not recommend.
To have some textbooks on linear algebra at hand is basically a good idea for one can always quickly look up forgotten things.

So you might have a look on your own or ask the folks in the science books section here on PF for recommendations: https://www.physicsforums.com/forums/science-and-math-textbooks.21/
I think there is also a list of books somewhere on this forum but I have forgotten where.

As linear algebra is rather basic, there should be found many videos of it on youtube, or look for the Khan academy courses.
But I would ask the guys here on PF first. They normally know pretty well what is suitable.
As you can see above, I'm a bit outdated on nowadays sources.

P.S.: There are almost for sure free online pdf available. I simply don't know them.

 

[Budana]

I salute and appreciate your help, fresh_42. I take my hat off to you.

 

[fresh_42]

I salute and appreciate your help, fresh_42. I take my hat off to you.

There's no need to. I make a lot of mistakes. That's sometimes embarrassing, but often helpful. You would be surprised on how much more you can learn from your mistakes (and the insights following their correction) than from a correct solution in the first place which you soon will forget.
Don't be afraid of them. Perfection is rare.
... forget this please, if it will happen to you that you work in a nuclear power plant one day ...

 

[Budana]

I have been working on this problem (trying to understand a topic in an advanced engineering textbook) for 4 days now. Today, you put an end to my 4-day-agony.
"see you" in other posts!

 

[fresh_42]

I have been working on this problem (trying to understand a topic in an advanced engineering textbook) for 4 days now. Today, you put an end to my 4-day-agony.
"see you" in other posts!

I remember a line in a text we had to prepare for a seminar. There was a formula, then "obviously .." and another formula.
It took two of us three days and four substitutions on the representation of complex numbers and about 20 lines of calculation before it has been obvious to us, too.

http://www.smart-words.org/humor-jokes/language-humor/research-phrases-meaning.html