Prove that the system a , b , d , e , f , g , h , i > 0

Ayana Waller

Ayana Waller

Answered question

2022-06-03

Prove that the system
a , b , d , e , f , g , h , i > 0
a e + a i b d + e i f h = 0
a e i h f a b d i g b f = 0
is inconsistent.

Answer & Explanation

Karson Stokes

Karson Stokes

Beginner2022-06-04Added 4 answers

From the third equation, you know that a e i = h f a + b d i + g b f. Now set x = b d i + g b f.
Therefore
a e i = h f a + x
e i = h f + x a
e i > h f
The same technique can be used to find that
a e > b d
Now set y = e i h f and z = a e b d. These are both clearly positive. Now, rewriting the second equation using these variables, we get
a i + y + z = 0
Since all three terms on the left are greater than 1, this cannot be true.

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