The ideal gas law, represented by the equation PV = nRT, is a fundamental principle in chemistry that relates pressure (P), volume (V), number of moles (n), the ideal gas constant (R), and temperature (T). When faced with problems involving two sets of values for these variables, we can rearrange the ideal gas law to derive new equations that help us analyze the situation effectively.

For instance, if we have two pressures (P₁ and P₂), two volumes (V₁ and V₂), or two temperatures (T₁ and T₂), we can utilize the relationships established by the ideal gas law to find unknown variables. This approach is particularly useful in scenarios where conditions change, allowing us to maintain the integrity of the gas law while adapting to the new parameters.

In these cases, the equations can be rearranged as follows:

1. For pressure and volume: P₁V₁ = P₂V₂

2. For temperature and volume: (P₁V₁)/T₁ = (P₂V₂)/T₂

3. For moles and temperature: (n₁T₁)/P₁ = (n₂T₂)/P₂

These rearrangements allow for a systematic approach to solving problems involving gases under varying conditions, ensuring that the ideal gas law remains applicable. As we proceed to practical examples, these concepts will become clearer and more intuitive.