I have the following three equations:
$$r- q\delta_1 - q\delta_2 - q\delta_3 + p\delta_1\delta_2 + p\delta_1\delta_3 + p\delta_2\delta_3 - N\delta_1\delta_2\delta_3=0\tag{1}$$
$$s- r\delta_1 - r\delta_2 - r\delta_3 + q\delta_1\delta_2 + q\delta_1\delta_3 + q\delta_2\delta_3 - p\delta_1\delta_2\delta_3=0\tag{2}$$
$$t- s\delta_1 - s\delta_2 - s\delta_3 + r\delta_1\delta_2 + r\delta_1\delta_3 + r\delta_2\delta_3 - q\delta_1\delta_2\delta_3=0\tag{3}$$
$p$, $q$, $r$, $s$, $t$ are arbitrary real numbers. $N$ is an integer.
$\delta_1$, $\delta_2$, and $\delta_3$ are the unknowns.
Is there a method using matrices (or otherwise) that allows derivation of algebraic expressions for $\delta_1$, $\delta_2$, and $\delta_3$ using $p$, $q$, $r$, $s$, $t$, and $N$?
I know that there is a solution, and all* $\delta_n$ are real numbers.
*Maybe not all, but at least three.
My attempt for deriving an expression for $\delta_1$ from $(1)$, plugging it into $(2)$, deriving $\delta_2$ and plugging it into $(3)$ will most likely work, but very fast it becomes an arithmetic nightmare and I am certain there is a simpler way.
You can first solve the linear equations in the variables $$ x = \delta_1 + \delta_2 + \delta_3\\ y = \delta_1\delta_2 + \delta_1\delta_3 + \delta_2\delta_3\\ z = \delta_1 \delta_2 \delta_3 $$ so that $\delta_i$ will be the roots of the degree 3 polynomial $$ \lambda^3 - x\lambda^2 + y\lambda-z=0. $$