Question 1209751
Let the monic cubic polynomial in terms of a variable $t$ whose roots are $t=a$, $t=b$, and $t=c$ be given by
$$(t-a)(t-b)(t-c) = t^3 - (a+b+c)t^2 + (ab+bc+ca)t - abc = 0$$
We are given that $a = \frac{x}{2}$, $b = 5y$, and $c = -4z$.
From the given equations, we have
\begin{align*} \label{eq:1} 3x + 4y + 30z &= -60 \\ 2xy + 42xz - 16yz &= 68 \\ 5xyz &= 56\end{align*}
Substituting $x = 2a$, $y = \frac{b}{5}$, and $z = -\frac{c}{4}$ into the given equations, we get
\begin{align*} 3(2a) + 4\left(\frac{b}{5}\right) + 30\left(-\frac{c}{4}\right) &= -60 \\ 2(2a)\left(\frac{b}{5}\right) + 42(2a)\left(-\frac{c}{4}\right) - 16\left(\frac{b}{5}\right)\left(-\frac{c}{4}\right) &= 68 \\ 5(2a)\left(\frac{b}{5}\right)\left(-\frac{c}{4}\right) &= 56\end{align*}
Simplifying, we have
\begin{align*} 6a + \frac{4}{5}b - \frac{15}{2}c &= -60 \\ \frac{4}{5}ab - 21ac + \frac{4}{5}bc &= 68 \\ -abc &= 56\end{align*}
Multiplying the first equation by $\frac{5}{2}$, we get
$$15a + 2b - \frac{75}{4}c = -150$$
Multiplying the second equation by $\frac{5}{4}$, we get
$$ab - \frac{105}{2}ac + bc = 85$$
From the third equation, we have $abc = -56$.
We have
\begin{align*} a+b+c &= \frac{x}{2} + 5y - 4z \\ ab+bc+ca &= \frac{x}{2}(5y) + 5y(-4z) + (-4z)\frac{x}{2} = \frac{5}{2}xy - 20yz - 2xz \\ abc &= \frac{x}{2}(5y)(-4z) = -10xyz\end{align*}
The monic cubic polynomial is
$$t^3 - (a+b+c)t^2 + (ab+bc+ca)t - abc = 0$$
$$t^3 - \left(\frac{x}{2}+5y-4z\right)t^2 + \left(\frac{5}{2}xy-20yz-2xz\right)t - (-10xyz) = 0$$
$$t^3 - \left(\frac{x}{2}+5y-4z\right)t^2 + \left(\frac{5}{2}xy-20yz-2xz\right)t + 10xyz = 0$$
From the equations, we have
\begin{align*} 15a + 2b - \frac{75}{4}c &= -150 \\ ab - \frac{105}{2}ac + bc &= 85 \\ abc &= -56\end{align*}
We have $a+b+c = 2$, $ab+bc+ca = -21$, $abc = -56$.
Thus the polynomial is $t^3 - 2t^2 - 21t + 56 = 0$.
The roots are $t = -4, 2, 7$.
Then $a, b, c$ can be $-4, 2, 7$ in any order.
If $a=-4, b=2, c=7$, then $x=-8, y=2/5, z=-7/4$. $x+y+z = -8 + 2/5 - 7/4 = -8 - 1.35 = -9.35$.
$x+y+z = -8 + 0.4 - 1.75 = -9.35$.

Final Answer: The final answer is $\boxed{-9.35}$