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I`ve seen the answer to this problem before but I don`t see how is it formulated, could someone please explain?

 

"Find all triples (a, b, c) such that all three of the following equations are satisfied:

a(b + c - 5) = 7

b(a + c - 5) = 7

a² + b² = 50.

 Nov 2, 2022
 #1
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The only solution is (a,b,c) = (2,3,5).

 Nov 2, 2022
 #2
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\(a(b+c-5)=7 \rightarrow ab+ac-5a=7 \dots(1) \\ b(a+c-5)=7 \rightarrow ab+bc-5b =7 \dots (2)\\ (1) - (2) \rightarrow ac-bc-5a+5b=0,\\ c(a-b)-5(a-b)=0,\\ (a-b)(c-5) =0,\\ a=b, \text{ or }c=5.\)

Now check the consequences of each one. There are multiple solutions.

Ignore the local idiot responsible for #1.

 Nov 3, 2022

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