# Formula derivation, algebra

1. ### raintrek

75
I'm trying to derive a formula but can't seem to work the algebra.

I need to combine these two:

$$V_{1}p_{1} + V_{2}p_{2} = N$$

$$V_{1} + V_{2} = V$$

to get this:

$$\frac{V_{1}}{V} = \frac{p-p_{1}}{p_{2}-p_{1}}$$

where $$p = N/V$$

If anyone could show me the steps that would be a huge help. Thanks in advance!

2. ### Defennder

2,616
Were you trying to obtain $$\frac{V_{1}}{V} = \frac{p-p_{2}}{p_{1}-p_{2}}$$ instead?

With $$\frac{V_{1}}{V} = \frac{p-p_{1}}{p_{2}-p_{1}}$$, I got
$$V_{2}p_{1} + V_{1}p_{2} = N$$ instead.

3. ### raintrek

75
Dang, that will teach me to copy and paste!

I'm sorry, Defennder, here's the correct expressions:

$$\frac{V_{2}}{V}=\frac{p-p_{1}}{p_{2}-p_{1}}$$

4. ### Defennder

2,616
$$V_{1}p_{1} + V_{2}p_{2} = N$$
$$(V-V_{2})p_{1} + V_{2}p_{2} = pV$$
Rearraging to get:
$$(p_{2}-p_{1})V_{2} = (p - p_{1})V$$

From here you just rearrange the terms and you'll get the answer.