
Prove Results For Given Variation


For proportional (µ) sign, we use '@' sign for input. 



Prove Results For Given Variation

Here X^{2} = X^2 = X2 and 2X = 2*X
For proportional (µ) sign, we use '@' sign for input.
You can solve example like ...
X @ Y => X^{3} + Y^{3} @ X^{2}Y  XY^{2}
X @ Y => X^{2} + Y^{2} + XY @ X^{2}  Y^{2}
X @ Y, Y @ Z => X^{3} + Y^{3} + Z^{3} @ XYZ
Y @ Z,X @ Y => X^{3} + Y^{3} + Z^{3} @ XYZ
5X  7Y @ 6X + 3Y => X @ Y
5X + 2Y @ 3X  7Y => X @ Y
9X  3Y @ 4X + 5Y => X @ Y
2X^{2}  3Y^{2} @ 4X^{2} + 5Y^{2} => X @ Y
4X^{3} + 5Y^{3} @ 2X^{3}  9Y^{3} => X @ Y
X^{2} + 9Y^{2} @ XY => X @ Y
4X^{2} + 9Y^{2} @ XY => X^{2} + Y^{2} @ XY

Prove Results For Given Variation

1. If `X prop Y,` then prove that `X^3+Y^3 prop X^2YXY^2`
`X prop Y`
`=> X=M*Y` (where constant `M != 0`)
Now `(X^3+Y^3) / (X^2YXY^2)`
`= (M^3Y^3+Y^3) / (M^2Y^3MY^3)`
`= (Y^3(M^3+1)) / (MY^3(M1))`
`= ((M^3+1)) / (M(M1))`
`=` nonzero constant
`:. X^3+Y^3 prop X^2YXY^2`
2. If `5X7Y prop 6X+3Y,` then prove that `X prop Y`
`5X7Y prop 6X+3Y`
`=> 5X7Y = M(6X+3Y)` (where constant `M != 0`)
`=> 5X7Y = 6MX+3MY`
`=> 5X6MX = 7Y+3MY`
`=> X(56M) = Y(7+3M)`
`=> X/Y = (7+3M) / (56M)`
`=> X/Y = `nonzero constant
`=> X prop Y`



