Find the value of a + b + c + d Let $a$ and $b$ be the roots of the equation: $x^2 - 10cx - 11d = 0$ where $c$ and $d$ be the roots of $x^2 - 10ax - 11b = 0$. Find the value of $a+b+c+d$, assuming that they all are distinct.
I first tried making an equation with roots $(a+b)$ and $(c+d)$ to get the sum of the roots, however I wasn't able to solve this question using that method as the answer which I got was in terms of the variables itself.
I also tried placing $a$ into the first equation and $c$ into the second to cancel out a common term ($-10ac$), but after cancelling, I got: $(a^2 - c^2 - 11d + 11b = 0)$. Now I don't know how to move ahead.

We have
$a+b=10c$ 
and
$c+d=10a$, hence
$$b=10c-a$$
and
$$d=10a-c$$
We also have $ab=-11d$ and $cd=-11b$, hence
$$10ac-a^2=-11(10a-c)$$
and
$$10ac-c^2=-11(10c-a)$$
Subtracting these gives
$$a^2-c^2=-11(10c-a)+11(10a-c)=121(a-c)$$
Assuming $a\not=c$, we get
$$a+c=121$$
Going back to the formulas for $b$ and $d$, we have
$$b+d=(10c-a)+(10a-c)=9(a+c)$$
Thus
$$a+b+c+d=(a+c)+(b+d)=(a+c)+9(a+c)=10(a+c)=1210$$
