Exploring the Range of Equilibrium States in Tank Systems: A Ratio Approach

[MathewsMD]

Given two equilibrium equations for a tank 1 and tank 2 with $Q^E_1 =6(9q_1 +q_2)$ and $Q^E_2 =20(3q_1 +2q_2)$, respectively, where $q_1, q_2 ≧ 0$, describe which possible equilibrium states for various values of $q_1$ and $q_2$ are possible.

I believe I know how the answer was derived, but would like an explanation, if possible.

What was done was:

Take $\frac {Q^E_2}{Q^E_1}$ and then substitute $q_1 = 0$ to find one extrema, and then $q_2 = 0$ for another extrema. This yielded $\frac {10}{9} ≤ \frac {Q^E_2}{Q^E_1} ≤ \frac {20}{3}$. Now I understand the logic used somewhat (i.e. use the minimum values of q1 and q2 to to see where the maximum and minimum of the possible equilibria states lie), but why exactly is the ratio taken? Are not specific values for the equilibrium states wanted as per the question? How exactly does the ratio reveal the specific min and max for the equilibrium states? How do we know there is no higher or lower value for the equilibrium if $q_1, q_2 ≠ 0$?

I feel like I am missing something here and any clarification would be greatly appreciated!

 

[MathewsMD]

When just trying to "describe" the equilibrium states, is the ratio sufficient? Does the ratio have any particular meaning?

Also, is there a way to find the exact equilibrium states as opposed to a ratio?