关于含3个未知数的n元复杂方程组线性化替换规则的正确性及优化方法问询

Great question—let’s break this down clearly!

First off, you’re absolutely correct about that substitution pitfall. When you define a new unknown (like $X'$) as a function of known terms from the original equations, you’re essentially creating a one-to-many mapping between solutions of $X'$ and the original unknown $X$. That’s why you end up with $n$ solutions for $X$ per $X'$ solution—your substitution is introducing extra degrees of freedom or reversing dependencies in a way that blows up the solution set.

Now, for avoiding this mess, the key is to define new variables to directly replace the non-linear terms themselves, not to mix knowns and unknowns in your substitution. Let’s use your example to make this concrete:

Looking at your first equation:
$$
\begin{aligned}
X + a \cdot b^{\frac{1}{Y}} \cdot Z &= c \
\end{aligned}
$$
The non-linear part here is $b^{\frac{1}{Y}}$. Instead of trying to relate a new variable to knowns like $a$ or $c$, define a new variable specifically for that non-linear term: let’s say $W = b^{\frac{1}{Y}}$.

Now you can rewrite the first equation as a simpler form, and if you want full linearity, take it one step further: define $V = W \cdot Z = Z \cdot b^{\frac{1}{Y}}$. This turns the original non-linear equation into a fully linear one:
$$X + a \cdot V = c$$

When you solve the linear system for your new variables ($X$, $V$, $X'$, $Y'$, $Z'$, etc.), you can map back to the original unknowns without creating multiple solutions—because each new variable is directly tied to a single non-linear component of the original equation, not a messy mix of knowns and unknowns.

Another rule of thumb: always make sure your substitution is a one-to-one mapping (or at least has a well-defined inverse) between the original unknowns and the new ones, whenever possible. If you can invert the substitution to get back the original unknowns uniquely from the new ones, you won’t end up with extra solutions.

To clarify with your example again: if you had defined $X' = \frac{c - X}{a \cdot Z}$ (a function of knowns $a,c$ and unknowns $X,Z$), that’s a bad substitution because solving for $X$ gives $X = c - a \cdot Z \cdot X'$—and since $Z$ is also an unknown, this introduces ambiguity. But defining $W = b^{\frac{1}{Y}}$ gives you a clear inverse: $Y = \frac{1}{\log_b W}$ (as long as $W > 0$, which aligns with the original term), so each $W$ maps to exactly one $Y$, no extra solutions.

To sum up:

• Your initial observation is totally on point—avoid defining new variables as functions of knowns + original unknowns if you want to avoid multiple solution branches.
• Instead, target the non-linear terms directly with new variables, ensuring each substitution has a clear, unique inverse back to the original unknowns.

备注：内容来源于stack exchange，提问作者Es_a