I have three formulas to calculate dilutions of three chemicals ($x_1$, $x_2$, $x_3$):
$x_1 * 20 / (x_1 + x_2 + x_3) * 0.75 / 7 = 0.9$
$x_2 * 20 / (x_1 + x_2 + x_3) * 0.75 / 7 = 0.1$
$x_3 * 10 / (x_1 + x_2 + x_3) * 0.75 / 7 = 0.2$
I know, this is a homogeneous equation system. Since I am needing all chemicals, $x_1$, $x_2$, and $x_3$ must not be zero. To this end, I am looking for approximated results with $x_1 > 0$, $x_2 > 0$ and $x_3 > 0$.
Do you have any suggestions to solve this problem? Thanks a lot!
The system is not consistent and you want to reconcile it.
A way to do it is to define a norm $$\Phi(x_1,x_2,x_3)=\sum_{i=1}^3 w_i\big[\text{lhs}_i-\text{rhs}_i\big]^2$$ and to minimize it under positivity constraints.
Making the story short, with all $w_i=1$, this would give $$x_1=1 \qquad \qquad x_2=\frac{47}{215} \qquad \qquad x_3=\frac{188}{215}$$ For these values, the rhs are $\frac{43}{42}=1.024$, $\frac{47}{210}=0.224$ and $\frac{47}{105}=0.448$