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Problem 721

AIME late

Algebra Difficulty 5.3 Find the answer

32. Given $3 a-2|b|=5, 4|a|-6 a=3 b$ , then $a^{2}+b^{2}=$

Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.

Your answer

A number or a short expression. Spacing, $ signs and \frac vs / are all fine.

Official solution

Answer: 13.

Solution: From the given, we have
$3 a=2|b|+5>0,$

then
$3 b=4|a|-6 a=-2 a0, \quad b<0 \text {. }$

Thus, from the given equation, we get
$\left\{\begin{array}{l} 3 a+2 b=5 \\ -2 a=3 b \end{array}\right. \text {, }$

Solving, we get
$\left\{\begin{array}{l} a=3 \\ b=-2 \end{array},\right.$

Therefore,
$a^{2}+b^{2}=13 \text {. }$

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.