How to convert the given set of equations into a symmetric form How do I convert the equations
$$\left\{\begin{array}{} 3a +2b+ c & = & 0\\ a+4b+4c & = & 0 \end{array}\right.$$
to the form
$$\frac{a}{4}=\frac{b}{-11}=\frac{c}{10}=k.$$
Thank you

One way of doing this is to relate $\frac a4$ to $b$ and $c$, separately. 
First, multiplying the first equation by $2$ gives
$2\times (3a+2b+c)=2\times0 \Leftrightarrow 6a+4b+2c=0$.
Subtracting the second equation from this result gives
$5a-2c=0$
$5a=2c$
$a=\frac{2}{5}c$
$\frac{a}{4}=\frac{c}{10}$.
Similarly, to relate $\frac{a}{4}$ to $b$, you can start by multiplying the second equation by $3$, then subtracting it from the first equation to get
$(3a+2b+c)-(3a+12b+12c)=0$
$-10b-11c=0$
$-11c=10b$
$c=\frac{10b}{-11}$.
Dividing everything by $10$ gives
$\frac{c}{10}=\frac{10b}{-110}=\frac{b}{-11}$.
But we know, $\frac{a}{4}=\frac{c}{10}$, so $\frac{a}{4}=\frac{-b}{11}$.
This simplifies to 
$\frac{a}{4}=\frac{b}{-11}=\frac{c}{10}$,
as required. 
You can google "linear simultaneous equations" for more details on how to solve this kind of questions. 
