Simultaneous Equation (or is it?)

Solve simultaneously

$X+Y+Z=10$
$XYZ=30$
$XY+YZ+XZ=31$

We could attempt to solve this simultaneously, but I think the algebra would be tricky.

The three equations are related to the roots of a cubic polynomial.

If the general equation of the polynomial is $ax^3+bx^2+cx+d$ , then we know

The sum of the roots

The product of the roots

and

So from our three equations we have

(1) $\begin{equation*}X+Y+Z=10=\frac{-b}{a}\end{equation*}$

(2) $\begin{equation*}XYZ=30=\frac{-d}{a}\end{equation*}$

(3) $\begin{equation*}XY+YZ+XZ=31=\frac{c}{a}\end{equation*}$

Let $a=1$ , then $b=-10, c=31$ , and $d=-30$

Our cubic is $t^3-10t^2+31t-30$ and we can try to solve it.

The roots will be factors of $-30$ , so $\pm 1, \pm 2, \pm 3, \pm 5, \pm 6, \pm 10, \pm 15, \pm 30$

Try $t=2$

$\begin{equation*}(2)^3-10(2)^2+31(2)-30=0\end{equation}$

Hence $t=2$ is a root.

Use synthetic division to find the quadratic factor

The quadratic factor is $x^2-8x+15$ , which factorises to $(x-3)(x-5)$

Hence the solutions are $X=2, Y=3$ , and $Z=5$

We could assume the solutions are natural numbers, then we can look at factors of 30.

Hence the solutions are $X=2, Y=3,$ and $Z=5$

But with this approach we might not be able to find the solutions.

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