How Do You Isolate TF in the Method of Mixtures Equation?

[Petrucciowns]

How would you solve cl *ml *(TL-TF)= cg*mg*(TF-TG)

for TF?

The text gives it as

Tf= cl*ml*TL+cg*mg*Tg / cl*ml+cg*mg


If you are wondering the basic equation is for method of mixtures

Thank you.

 

[Doc Al]

First thing to do is multiply it out to get rid of the parentheses. Hint: a*(x - y) = a*x - a*y.

Then collect all the terms with the Tf factor to one side.

 

[Petrucciowns]

First thing to do is multiply it out to get rid of the parentheses. Hint: a*(x - y) = a*x - a*y.

Then collect all the terms with the Tf factor to one side.

Hmmm, thanks for the help but, I'm still not getting it. Here is what I'm trying so far. Sorry algebra has never been my strong point. :(


cl* ml*TL-ml*TF = cg* mg* TF - mg *TG

then TF/1= cg * mg* TF-mg*TG / cl *ml*TL-ml ----- I think this is where I screw it up


then: TF/TF = cg * mg -mg * TG / cl * ml* TL-ml


as you can see I'm probably doing this all wrong and am getting myself confused.

 

[berkeman]

Hmmm, thanks for the help but, I'm still not getting it. Here is what I'm trying so far. Sorry algebra has never been my strong point. :(


cl* ml*TL-ml*TF = cg* mg* TF - mg *TG

then TF/1= cg * mg* TF-mg*TG / cl *ml*TL-ml ----- I think this is where I screw it up


then: TF/TF = cg * mg -mg * TG / cl * ml* TL-ml


as you can see I'm probably doing this all wrong and am getting myself confused.

You lost some terms right off the bat there. Check your first line -- you didn't distribute the cl*ml across both terms in the parenthesis on the left-hand side (LHS). And you made the same error on the RHS.

 

[Doc Al]

No shortcuts: a*b*(x - y) = a*b*x - a*b*y

 

[Petrucciowns]

Soo... the first line would look more like:

cl*ml*TL- cl*ml*TF = cg*mg*TF-cg*mg*TG?

 

[Doc Al]

Soo... the first line would look more like:

cl*ml*TL- cl*ml*TF = cg*mg*TF-cg*mg*TG?

Now you're cooking!

 

[Petrucciowns]

lol..

So I'm probably going to screw up again ,but would the next line be TF = cg x mg x TF - cg x mg x TG / cl x ml x TL - cl x ml ?

 

[Doc Al]

So I'm probably going to screw up again ,but would the next line be TF = cg x mg x TF - cg x mg x TG / cl x ml x TL - cl x ml ?

Do the second step that I suggested: Starting with your new first line, move all the terms that contain TF to the left and all others to the right.

 

[Petrucciowns]

By doing that I get: TF= cg x mg x TG / Cl x ML x TL- CL x MLOnce again my algebra skills are really lacking so bare with me.

 

[Doc Al]

By doing that I get: TF= cg x mg x TG / Cl x ML x TL- CL x ML

Just move the terms to one side. Don't divide by anything yet. (One step at a time.)

 

[Petrucciowns]

But both sides are filled with variables, don't you have to clear one side to be able to have the side clear for TF?

 

[Doc Al]

But both sides are filled with variables, don't you have to clear one side to be able to have the side clear for TF?

No. In your equation in post #6 there are 4 terms (each with multiple factors, like a*b*c). Two of them contain Tf and two do not. Move things around so that the two Tf terms are on the left side and the others are on the right.

 

[Petrucciowns]

Ok, I see so cl*ml*TL- cl*ml*TF = cg*mg*TF-cg*mg*TG becomes:

cl*ml*TF+cg*mg*TF= cg*mg*TG+cl*ml*TL?

 

[Doc Al]

Ok, I see so cl*ml*TL- cl*ml*TF = cg*mg*TF-cg*mg*TG becomes:

cl*ml*TF+cg*mg*TF= cg*mg*TG+cl*ml*TL?

Looks good. Now you can take the left hand side and factor out the TF, adding in the correct parentheses. It will be wise to put parentheses around the right hand side too.

 

[Petrucciowns]

What do you mean like this?

(cl*ml)(TF)+(cg*mg)(TF)= (cg*mg)(TG)+(cl*ml)(TL)

then

TF/TF = TF= (cg*mg)(TG)+(cl*ml)(TL) / (cl*ml)+(cg*mg)Looks like I got it ehhhh? I really appreciate your help, I hope I can remember this

 

[Redbelly98]

What do you mean like this?

(cl*ml)(TF)+(cg*mg)(TF)= (cg*mg)(TG)+(cl*ml)(TL)

then

TF/TF = TF= (cg*mg)(TG)+(cl*ml)(TL) / (cl*ml)+(cg*mg)


Looks like I got it ehhhh? I really appreciate your help, I hope I can remember this

I think you have the right idea in your head, but you have written things down rather sloppily ... in essence, wrongly.

For starters,

TF/TF = TF​

is simply not true. Forget the TF/TF here.

TF= (cg*mg)(TG)+(cl*ml)(TL) / (cl*ml)+(cg*mg)

Can you add parantheses, in two places, to make this a correct statement? Maybe use square parantheses [], in order not to confuse distinguish them with the regular () kind.

As written, this equation means

TF = [(cg*mg)(TG)] + [(cl*ml)(TL) / (cl*ml)] + [(cg*mg)]​

which probably isn't what you meant.

 

[Doc Al]

What do you mean like this?

(cl*ml)(TF)+(cg*mg)(TF)= (cg*mg)(TG)+(cl*ml)(TL)

then

TF/TF = TF= (cg*mg)(TG)+(cl*ml)(TL) / (cl*ml)+(cg*mg)


Looks like I got it ehhhh? I really appreciate your help, I hope I can remember this

As Redbelly98 says, it looks like you've got the right idea, but you need to be more careful when writing it out. Here's the outline of what I had in mind:

Starting with something like this:
a*X + b*X = c*Y + d*Z

Factor out the X, adding in parentheses as needed:
(a + b)*X = (c*Y + d*Z)

Solve for X and you're done.