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

Question

Weird title perhaps, so let me illustrate with the question that got me thinking about this problem:

You are buying a laptop and have two to choose from. What is the difference between the original prices of the two laptops?

What you know: After the laptops have come down in price 35 % and 45 % respectively, the difference in price between them is $50.

Can this answer be determined?

Intuitively, I would say no. If I write the problem algebraically I get:

0.65a - 0.55b = 50

It's obvious we can never work out a or b from this, but that's not what they're asking. They're asking what a - b (or rather |a - b|) is.

I understand that this answer still can't be determined, but if the percentages were 50 (or any same number) I could instead write:

0.5a - 0.5b = 50

Which simplifies to:

a - b = 100

I'm not quite understanding why I can't solve for a - b from any equation that includes a and b. What's the inutition to help me understand why this question is unsolvable in case of different percentages, but solvable in the case of the same?

Answer 1

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 $0.65a-0.55b$.

Answer 2

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.

Answer 3

Notice these are simultaneous equations. You can break $a$ and $b$ out using a thing called parametric equations.

Suppose that $a=bc$ for some $c$. There indeed must exist a $c$ for each $a,b$.

Then, we would have (original problem)

$$0.65bc-0.55b=50$$

$$b=\frac{50}{0.65c-0.55}$$

Similarly, if we multiply by $c$, we get $a=bc$, so

$$a=\frac{50c}{0.65c-0.55}$$

Now we can solve for $a-b$, which is

$$a-b=\frac{50(c-1)}{0.65c-0.55}$$

In the case that it was a 50/50 relationship, the denominator would have a factor of $c-1$, which would cancel and we'd have no $c$'s left over. However, this is not the case, and so we are left with this, our solution in terms of $c$.