Look at the difference in algebraic operation for the 2 math examples below:
2x + 4y - 3z = 3
after re-shuffling, becomes
-3z +2x + 4y - 3 = 0
3x - 4y + z = 2
becomes, after re-shuffling,
-z -4y -3z -2 = 0
Example A can be seen to be correct mathematically, whereas, Example B isn't.
Example B, after having the terms re-shuffled, has the signs of those terms changed!
The explanation to this sign change is that since, the terms were moved from left to right, and right to left, the sign must change. A shocking mistake has been made!
This is a mis-understanding and also a mis-conception.
What was missed out here is that the movement of left to right (or vice versa) has to cross over the "equal" symbol.
4x -5y = 1
0 = 1 - 4x + 5y <== This is correct sign change after re-shuffling across the "=" symbol.
-5y + 4x = 1 <== This is correct re-shuffled terms with no change in sign.
As long as the terms remain on the same side of the "=" symbol, the terms will not have their signs changed, even though their positions may have shifted.
The sign change results only when the term moves across the "equal" symbol, crossing over the opposite side.
Thus, do not confuse re-shuffling of terms within the same side to crossing the "equal" symbol.
This simple mistake can produce a big mistake through wrong understanding of math principles.
Math make us think properly and logically with reasoning to every steps taken. It is a good subject that aids mankind. Treasure the learning.