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Solve the system of equations $$\begin{cases} \sqrt{x+2y+3}+\sqrt{9 x+10y+11}=10,&\\[10pt] \sqrt{12 x+13y+14}+\sqrt{28 x+29y+30}=20. \end{cases} $$

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Hint: Try $t=x+y+1$ to get

\begin{cases} \sqrt{t+y+2}+\sqrt{9t +y+2}=10,&\\[10pt] \sqrt{12t+y+2}+\sqrt{28t+y+2}=20. \end{cases}

And solve by squaring.

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In this sort of question, repeated squaring is involved. So any big numbers that you start with (like $28$) are going to produce much bigger numbers, and multiple-term sums will blow up unpleasantly. On the other hand, linear substitutions (such as Maazul has suggested) are easily reversed at the end. Substituting $u=t+y+2$ as a further step in Maazul's method reduces the number of terms significantly, although leaving the $28$ diminished only to $27$. So I suggest $4u=4t+y+2$ instead. While this worsens the first equation somewhat, it keeps the lid on the numbers overall since, after dividing the second equation through by $2$, the biggest number that appears in the resulting equations in $t$ and $u$ is $10$.

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