How to get an expression for $\frac{e1}r$ out of these four equations?

Question

Given the following equations:

$$y = g_2 \cdot e_3 \\ e_3 = e_2 - y \cdot h_2 \\ e_2 = g_1 \cdot e_1 \\ e_1 = r - h_1 \cdot e_2 - h_3 \cdot y$$

How can I get an expression of $\dfrac{e1}r$ in terms of $g_1$, $g_2$, $g_3$, $h_1$, $h_2$ and $h_3$?

Answer 1

Use the first and third equations, replace $y$ and $e_2$ in the other two equations. You now have two equations in two unknowns $e_1$ and $e_3$.

Answer 2

So, let's look at $e_{1}$ and substitute in all the others, in sequence. \begin{eqnarray} e_{1} &=& r-h_{1}.e_{2}-h_{3}.y \\ &=& r-h_{1}e_{1}g_{1}-h_{3}g_{2}e_{3} \\ &=& r-h_{1}e_{1}g_{1}-h_{3}g_{2}(e_{2}-yh_{2}) \\ &=& r-h_{1}e_{1}g_{1}-h_{3}g_{2}e_{2}-yh_{3}g_{2}h_{2} \\ &=& r-h_{1}e_{1}g_{1}-h_{3}g_{2}g_{1}e_{1}-yh_{3}g_{2}h_{2} \end{eqnarray} Now just collect like terms in $e_{1}$ and re-arrange to the form you need.