<s>[INST] Solving a three variable equation I have three given values, suppose a=1.86, b=2.6 and c=4.2. Now I have to figure out x,y,z such that 


*

*$x\gt 0,y\gt 0$ and $z\gt 0$   

*$x+y+z=1$  

*$a*x\gt 1, b*y\gt 1$ and $cz\gt 1$  
I need a generalized solution steps for this to implement in programming.
Thanks.
 [/INST] $\left\{ \begin{array}{l}
x > \frac{1}{a} \Rightarrow x = \frac{1}{a} + \varepsilon \\
y > \frac{1}{b} \Rightarrow y = \frac{1}{b} + \varepsilon \\
z > \frac{1}{c} \Rightarrow z = \frac{1}{c} + \varepsilon 
\end{array} \right.$ where $0<\varepsilon$.
$\Rightarrow \frac{1}{a} + \varepsilon  + \frac{1}{b} + \varepsilon  + \frac{1}{c} + \varepsilon  = 1 \Rightarrow \varepsilon  = \frac{1}{3} - \frac{1}{{3a}} - \frac{1}{{3b}} - \frac{1}{{3c}} = {\rm{0}}{\rm{.1053}}$
</s>