# Solve equation to find value

#### LibbyF

##### New member
If I wanted to find the value of Cin for the following equation:

Cin = a P Cout – (ak) Cin + Q / V

Where Cin and Cout are the indoor and outdoor concentrations and the other parameters are as follows;

Cout = 15

P = 1

a = 1.2

k = 1.19

Q = 1.4

V = 2.5

How would I rearrange the equation to get the value of Cin?

#### Subhotosh Khan

##### Super Moderator
Staff member
If I wanted to find the value of Cin for the following equation:

Cin = a P Cout – (ak) Cin + Q / V

Where Cin and Cout are the indoor and outdoor concentrations and the other parameters are as follows;

Cout = 15

P = 1

a = 1.2

k = 1.19

Q = 1.4

V = 2.5

How would I rearrange the equation to get the value of Cin?
Suppose the equation was:

x = A - (B - C) * x + D

How would you rearrange the equation to get the value of 'x'?

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

Please share your work/thoughts about this problem.

#### Otis

##### Elite Member
Hi Libby. Have you substituted the known values and done any arithmetic?

That is, have you evaluated expressions a*P*Cout, a-k, and Q/V yet?

#### LibbyF

##### New member
I have evaluated the expressions as

Cin = (1.2x1x15) - (1.2-1.19) * Cin + (1.4/2.5)

Cin = (18 - 0.01) * (Cin + 0.56)

But then I don't know where to go from there.

In response to the earlier reply, for x = A - (B - C) * x + D I would normally try to get the x's on both sides by subtraction (like for a two-step equation), but I get stuck when there is only one x to play with.

#### Subhotosh Khan

##### Super Moderator
Staff member
Cin = (1.2x1x15) - (1.2-1.19) * Cin + (1.4/2.5)
Cin = 18 - 0.01 * Cin + 0.56

Cin * 1.01 = 18.56

Continue.....

#### Subhotosh Khan

##### Super Moderator
Staff member
Cin = 18.56 / 1.01 ?

Cin = 18.37 ?
Those two lines are correct. Why do you have doubt in your work?

#### LibbyF

##### New member
I guess I am a late bloomer with this stuff.

Does this mean I could rearrange the original equation;

Cin = a P Cout – (ak) Cin + Q / V

To

Cin - (a - k) = a P Cout + Q / V ?

#### Otis

##### Elite Member
I could rearrange the original equation To

Cin - (a - k) = a P Cout + Q / V ?
Almost. Had you added the term (a-k)*Cin to both sides?

If so, then the left-hand side ought to be Cin + (a-k)*Cin.

That can be simplified further (by factoring out Cin). You'll then be in a position to solve the equation for Cin (dividing each side by the resulting coefficient expression on Cin).

Let us know, if you need more help understanding the process (or vocabulary). Cheers

#### LibbyF

##### New member
I think I am a little confused about how to get from

Cin = 18 - 0.01 * Cin + 0.56

to

Cin * 1.01 = 18.56

If you add (- 0.01 * Cin) to both sides, do you not end with 2Cin ?

#### Otis

##### Elite Member
Cin = 18 - 0.01 * Cin + 0.56

If you add (- 0.01 * Cin) to both sides, do you not end with 2Cin ?
Hi Libby. Yes, we do not end up with 2Cin.

Cin = 18 - 0.01 * Cin + 0.56

When we look at that equation, we see two Cin terms. The term on the right side is the variable Cin multiplied by -0.01 (the coefficient).

In order to combine Cin terms, we add 0.01*Cin to each side because that will eliminate the Cin term on the right side.

0.01*Cin - 0.01*Cin = Cin(0.01 - 0.01) = Cin(0) = 0

What do you get on the left side?

#### LibbyF

##### New member
Cin = 18 - 0.01 * Cin + 0.56

Cin + 0.01 * Cin = 18.56

Cin(0.01 + 1) = 18.56

Cin * 1.01 = 18.56

Cin = 18.56 / 1.01

Cin = 18.37

#### LibbyF

##### New member
If I wanted to solve it dynamically (not sure if thats the right terminology) - by feeding the original equation a range of values and seeing how it affects Cin, could I rearrange to:

#### Otis

##### Elite Member
If I wanted to solve [the equation symbolically] could I rearrange [as shown]
Yes, your formula for Cin is correct. That's what I described in post #8. Good job.

#### HallsofIvy

##### Elite Member
If I wanted to find the value of Cin for the following equation:

Cin = a P Cout – (ak) Cin + Q / V

Where Cin and Cout are the indoor and outdoor concentrations and the other parameters are as follows;

Cout = 15

P = 1

a = 1.2

k = 1.19

Q = 1.4

V = 2.5

How
C_{in}\$ occurs in two places in the equation
$$\displaystyle C_{in}$$ occurs in two places in the equation
$$\displaystyle C_{in}= a P C_{out}- (a- k)C_{in}+ Q/V$$
so put them together- add $$\displaystyle (a- k)C_{in}$$ to both sides:
$$\displaystyle C_{in}+ (a- k)C_{in}=a P C_{out}+ Q/V\(\displaystyle Factor out \(\displaystyle C_{in}$$ on the left
$$\displaystyle (1+ a- k)C_{in}= a P C_{out}+Q/V$$

Divide both sides by 1+ a- k
$$\displaystyle C_{in}= \frac{a P C_{out}+ Q/V}{1+a- k}$$\)\)

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