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Solving a system of equations in W|A

Posted 10 months ago
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I am a new user and I just want to be pointed in the right direction.

I have a system of equations

P=Gt, F=fcG, G=D+F/(1-f)+U, T=P-F, U=(1-f)G, N=D-T+F

I want to know how to solve for N in terms of

t,f,c,D

I tried inputting

Solve P=Gt, F=fcG, G=D+F/(1-f)+U, T=P-F, U=(1-f)G, N=D-T+F for N,t,f,c,D

But Wolfram|Alpha said it didn’t understand.

Any assistance would be much appreciated.

Paul Mason

8 Replies
Posted 10 months ago

Try this with Wolfram|Alpha

 Solve[{P=G*t, F=f*c*G, G=D+F/(1-f)+U, T=P-F, U=(1-f)*G, N=D-T+F},{N,t,f,c,D}]

That worked. I posted this. Then I tried to check it again and it failed.

Working on finding the problem

Posted 10 months ago

Thank you Bill. I tried pasting your expression into Wolfram|Alpha. It just said it didn’t understand.

Do you know if there is a general reference in Wolfram|Alpha Documentation about how to pose this type of question?

Posted 10 months ago

Yes. That is what I'm seeing now too.

I'm almost certain it worked once and I can't get it to do that again.

If I do

solve P=G*t,F=f*c*G,G=D+F/(1-f)+U,T=P-F,U=(1-f)*G,N=D-T+F for N

it says it does not understand

But if I do

solve F=f*c*G,G=D+F/(1-f)+U,T=P-F,U=(1-f)*G,N=D-T+F for N

then it works.

There are limits on how long a line of input it will accept and it seems that there may be limits on how complicated internal computations may be. I do not know that is what is causing the problem at the moment.

Sometimes it is possible to divide a problem into two parts, each of which is small and simple enough to complete, and then combine the results to get your final answer

I just tried this

Solve[{P=G*t, F=f*c*G, G=D+F/(1-f)+U, T=P-F, U=(1-f)*G, N=D-T+F},N]

and it worked! And I closed the tab on the browser, opened again, went to Wolfram|Alpha, pasted that, <enter> and it worked again. See if you can reproduce that. I'm still not certain if the form of the result is what you are looking for. I'll look at this a little more and see if I can get it to do better.

Posted 10 months ago

Thank again Bill.

I copied in the version you got to work, to no avail.

I will check the syntax of my expressions; also try to break the problem up.

Paul

Posted 10 months ago

I tried a different approach

ReplaceAll[N=D-T+F, T->P-F]

gives me

N=D+2F-P

and then

ReplaceAll[N=D+2F-P, P->G*t]

gives me

N=D+2*f-G*t

and then

ReplaceAll[N=D+2*f-G*t, G->D+F/(1-f)+U]

gives me

N=-t(D+F/(1-f)+U)+D+2*f

What that is trying to do is eliminate one variable at a time and get closer to your desired expression only in terms of t,f,c,D

BUT you have things like

G=D+F/(1-f)+U
F=fcG

where G gives you something containing F and F gives you something containing G. Likewise G and U. What I had hoped was I might be able to order the replacements in a way to get a result only in terms of what you wanted.

But each of these ReplaceAll seems to work

Posted 10 months ago

Thanks again, Bill.

In a little while, I am going to try manually eliminating the intermediate variables I don’t want, then use Simplify.

A reference manual for Wolfram|Alpha would be really useful.

Paul

Posted 10 months ago

First let's see if we can make F, G and U independent of each other by using Solve

Solve[{F=f*c*G, G=D+F/(1-f)+U, U=(1-f)*G},{F,G,U}]

That neatly separates the F, G and U and gives us, among a couple of other things,

F=c*D*(f-1)/(c+f-1)
G=D*(f-1)/(f*(c+f-1))
U=-D*(f-1)^2/(f*(c+f-1))

We are very fortunate that all those are expressed in terms of the variables you are looking for.

Next eliminate the T in your original equation N=D-T+F using your given T=P-F

ReplaceAll[N=D-T+F, T->P-F]

which gives us

N=D+2*F-P

and then eliminate the P in that using your given P=G*t

ReplaceAll[N=D+2*F-P, P->G*t]

which gives us

N=D+2*F-G*t

Now eliminate the F in that using the result from Solve that we did above

ReplaceAll[N=D+2*F-G*t, F->c*D*(f-1)/(c+f-1)]

which gives us

N=2*c*D*(f-1)/(c+f-1)+D-G*t

and then eliminate the G in that using the result from Solve that we did above

ReplaceAll[N=2*c*D*(f-1)/(c+f-1)+D-G*t, G->D*(f-1)/(f*(c+f-1))]

which finally gives us

N=-D*(f-1)*t/(f*(c+f-1))+2*c*D*(f-1)/(c+f-1)+D

It seems like there should be a simpler and less error prone way of doing this where Wolfram|Alpha does more of the work in a single step. Perhaps someone else can see a way to do that or perhaps in a day or two you or I might realize a way to do that.

Please check all this VERY carefully to make certain I haven't made any mistakes.

Posted 10 months ago

Thank you again, Bill. My head is spinning now!

I will have a look at it later, probably tomorrow.

Paul

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