Finding the minimum sum

All numbers considered are positive real.

Given:

M spaces exist

In each space, N sets of the form (a,b,c) exist.

Now we have the following equations: \[({a_{11}} + {b_{11}} + {c_{11}}) + ({a_{21}} + {b_{21}} + {c_{21}}) + ... + ({a_{N1}} + {b_{N1}} + {c_{N1}}) = {S_1}\] for space 1 … … $({a_{1M}} + {b_{1M}} + {c_{1M}}) + ({a_{2M}} + {b_{2M}} + {c_{2M}}) + ... + ({a_{NM}} + {b_{NM}} + {c_{NM}}) = {S_M}$ for space M

We know that the sum $S_Y$ is minimum. Then can we say that if we deduct a constant x from each middle element of each set of each space, then for the same combination of sets, we will get the minimum value? I.e. the new $S_{Y}$ will remain the minimum sum?

The new equations after deduction are: $({a_{11}} + {b_{11}} - x + {c_{11}}) + ({a_{21}} + {b_{21}} - x + {c_{21}}) + ... + ({a_{N1}} + {b_{N1}} - x + {c_{N1}}) = {S_1}$ … … $({a_{1M}} + {b_{1M}} - x + {c_{1M}}) + ({a_{2M}} + {b_{2M}} - x + {c_{2M}}) + ... + ({a_{NM}} + {b_{NM}} - x + {c_{NM}}) = {S_M}$

#Algebra

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Comments

I suppose the answer is obviously yes. Remove the $x$ from the brackets, and you would get $nx$ . Since $S_Y\ge S_{\text{random}}$ , $S_Y-nx\ge S_{\text{random}}-nx$ wouldn’t it? :)

Jeff Giff - 3 months, 2 weeks ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$