How to check if a function doesn't depend on a variable?

[kelly0303]

Hello! I have some experimental data points $(z_i,dz_i)$ and I know that in the most general case this variable can be written in terms of 2 other variables as $z_i = ay_i+bx_i$. Beside $z_i$ I can also measure, for each point, $x_i$ (we can assume that the uncertainty in $x_i$ is negligible), but not $y_i$. I suspect, based on some calculations, that (at least at the level of the experimental uncertainties, $dz_i$) the $bx_i$ term will be negligible i.e. $b\sim 0$ given my uncertainties. Is there a way to test this experimentally, given my current data and the expected functional form? Thank you!

 

[FactChecker]

An important question is whether the $x_i$s and $y_i$s are independent or are correlated. If they are independent, then you can consider whether $b \approx 0$ without regard to the value of $a$. But if they are related, you must consider the value of $a$ to understand whether the addition of a nonzero $b$ is beneficial.
Since you can not measure the $y_i$s, I am afraid that the best you can do is to use regression to determine if the model $z_i = a x_i$ has a statistically significant non-zero $a$.

 

[WWGD]

In Calculus, the Inverse/Implicit function theorems are usually used with this purpose, when given an expression in terms of $x,y$.