An equation with two variables is unsolvable for either one, but how can I know if it's unsolvable as an expression for both?

As you said, we have: $$0.65a - 0.55b = 50$$ Let's try to guess and check different $(a, b)$ to solve this problem.

• Guessing $b=10$ gives us $a \approx \$85.38$, meaning $\lvert a-b \rvert \approx \$75.38$
• Guessing $b=20$ gives us $a \approx \$93.85$, meaning $\lvert a-b \rvert \approx \$73.85$

Thus, there are multiple possible values of $\lvert a-b \rvert$, so we can't solve for $\lvert a-b \rvert$ from the original equation: There simply isn't one unique answer for that expression we can solve for using only that equation.

The intuitive reason for why this is can be found using linear algebra. The equation at the top gives us the following augmented matrix: $$[\begin{matrix}0.65 \ -0.55 \ | \ 50\end{matrix}]$$ We want to solve for $a-b$ which can be represented as: $$[\begin{matrix}1 \ -1\end{matrix}]$$ However, $(0.65 \ -0.55)$ has a different ratio of elements than the row $(1 \ -1)$, so they're linearly independent. That means the first matrix can't be reduced to the second matrix, so you can't solve for $a-b$ in terms of $0.65a-0.55b$.

I believe its because the difference between $a$ and $b$ is dependent on their values in every case and it just so happens the case where the difference did not depend on them is when they both drop by the same percentage. Let me illustrate using the example you've given. $$0.65a - 0.55b = 50$$ $$0.55a - 0.55b = 50 - 0.10a$$ $$a-b = \frac{50-0.10a}{0.55}$$ As you can see the difference now depends on $a$ and this is generally what happens when the percentage drops are not equal. Intuitively think about it this way, if $a$ and $b$ both drop by a particular same percentage the difference will aswell, but if they don't the difference depends on how much of an impact the dicount on $a$ does with respect to $b$ which depends on their values.