possibility of unique value for constants

Question

If the equation $$(p+2q)x+(2p+5q)y=4a+3b $$ represents the equation of the line $$2x+3y=13$$ can the value of $p$ and $q$ be uniquely determined? No information about $a$ and $b$ has been provided. I feel that it cannot be uniquely determined as the equation of the line can also be written with all coefficients multiplied by some number which will give different values of $p$ and $q$. Is this reasoning correct?

Answer 1

You can determine $p$ and $q$ if you know $4a+3b$. But in the absence of such information, there are multiple solutions: one can multiply $p$, $q$, and $4a+3b$ by the same constant.

With some normalizing condition, such as $p+q=1$, the solution would be unique.

(Compilation of comments by Gerry Myerson)