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I can't see how to make this problem simpler so I can avoid fractions or big number arithmetic. I understand we can always find a $p$ and $q$ to satisfy either equation independently but because but coefficients are $1$ apart and the equations totaling the same value, I'm suspicious if there is actually a solution.

$\begin{align} 13p - 92q = 273 \\ 12p - 91q = 273 \\ \end{align}$

I'm trying to solve this using substitution.

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    $\begingroup$ one may subtract equations $\endgroup$
    – Will Jagy
    Mar 27, 2021 at 0:02
  • $\begingroup$ So use elimination? Ok, I will look at that. $\endgroup$
    – user839943
    Mar 27, 2021 at 0:06

1 Answer 1

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Well, you can always subtract equations $$ \begin{align*} 13p - 92q = 273 \\ 12p - 91q = 273 \\ \end{align*} $$ to discover that $$p - q = 0 \Longrightarrow p = q$$ and so by substituting one for the other we can find that $$ 13p - 92q = -79p = 273 \Longrightarrow p = q = \frac{-273}{79} $$

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