Microeconomics competitive equilibrium interest rate determination

Question

I've got a microeconomics question that involves rearranging an equation with summation, where the only constant are $1$ and $r$.

Firstly this is the equation stating that across individuals $i=1$ to $n$ demand for period 1 consumption $(x_1)$ should equal the supply of period 1 money $(e_1)$.

$\Sigma_{i=1}^nx_{i1}$($r^*$)=$\Sigma_{i=1}^n$$e_{i1}$ $[1]$

So this states the total demand for money across all individuals is equal to the supply of money in period 1 when the interest rate is optimum, my question is how would you rearrange this equation when the demand function for individual $i$ is substituted in to this equation so when:

$x_{i1}$($r^*$)=$e_{i1}(1+r)+e_{i2}\over(1+r)(1+\beta_i$

So substituting this derived demand function can be plugged into equation $[1]$ gives this:

$\sum_{i=1}^n$$e_{i1}(1+r)+e_{i2}\over(1+r)(1+\beta_i)$$=\Sigma_{i=1}^n e_{i1}$

Ignoring the economic details how do you just make $(1+r)$ the subject of this equation?

Answer 1

Start by dividing. Note that $$\frac{e_{i1}(1+r)+e_{i2}}{(1+r)(1+\beta_i)}=\frac{e_{i1}(1+r)}{(1+r)(1+\beta_i)}+\frac{e_{i2}}{(1+r)(1+\beta_i)} = \frac{e_{i1}}{1+\beta_i}+\frac{e_{i2}}{(1+r)(1+\beta_i)} .$$ It follows that $$\sum_1^n \frac{e_{i1}}{1+\beta_i}+\frac{1}{1+r}\sum_1^n \frac{e_{i2}}{1+\beta_i}=\sum_1^n e_{i1}.$$ This has the shape $$A+\frac{1}{1+r}B=C.$$ So $\frac{1}{1+r}=\frac{C-A}{B}$. Now it's almost finished.