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I have the following equations

$$\begin{aligned} - \frac { a } { C _ { 1 } } Q _ { 1 } + \frac { 1 - a } { H _ { 1 } } + \beta \left( \frac { a } { C _ { 2 } } \frac { \partial C _ { 2 } } { \partial H _ { 1 } } + \frac { 1 - a } { H _ { 2 } } \frac { \partial H _ { 2 } } { \partial H _ { 1 } } \right) + \mu s \frac { Q _ { 2 } } { R } & = 0 \\ \frac { a } { C _ { 1 } } + \beta \left( \frac { a } { C _ { 2 } } \frac { \partial C _ { 2 } } { \partial B _ { 1 } } + \frac { 1 - a } { H _ { 2 } } \frac { \partial H _ { 2 } } { \partial B _ { 1 } } \right) - \mu & = 0 \end{aligned}$$

The partial derivatives are given as:

$$\frac { \partial C _ { 2 } } { \partial H _ { 1 } } = a Q _ { 2 } , \quad \frac { \partial H _ { 2 } } { \partial H _ { 1 } } = 1 - a , \quad \frac { \partial C _ { 2 } } { \partial B _ { 1 } } = - R a , \quad \frac { \partial H _ { 2 } } { \partial B _ { 1 } } = - \frac { R ( 1 - a ) } { Q _ { 2 } }$$

Moreover,

$$\begin{aligned} C _ { 2 } & = a \left( Y - R B _ { 1 } + Q _ { 2 } H _ { 1 } \right) \\ H _ { 2 } Q _ { 2 } & = ( 1 - a ) \left( Y - R B _ { 1 } + Q _ { 2 } H _ { 1 } \right) \end{aligned}$$

With the above in mind, I need to arrive at a consolidated equation (i.e. adding the two first equations)

$$\frac { 1 - a } { H _ { 1 } } + \frac { \beta \left( Q _ { 2 } - Q _ { 1 } R \right) } { Y - R B _ { 1 } + Q _ { 2 } H _ { 1 } } = \mu \left( Q _ { 1 } - s \frac { Q _ { 2 } } { R } \right)$$

I've tried a couple of times but cannot seem to arrive at the desired equation. Any help is highly appreciated :-)

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  • $\begingroup$ perhaps multiplying both sides of the second equation by $Q_1$ will allow you to progress to the next step. $\endgroup$
    – John Joy
    Commented Jan 12, 2019 at 14:07
  • $\begingroup$ I have given that a go, however I end up with a (1-a)/H_1 *Q_1 term I cannot seem to get rid of (or alternatively, a Q_1 on both mu's that shouldnt be there) :-). $\endgroup$
    – user469216
    Commented Jan 13, 2019 at 12:16

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