1. If `x prop y,` then prove that `x^3+y^3 prop x^2y-xy^2`
`x prop y`
`=> x=M*y` (where constant `M != 0`)
Now `(x^3+y^3) / (x^2y-xy^2)`
`= (M^3y^3+y^3) / (M^2y^3-My^3)`
`= (y^3(M^3+1)) / (My^3(M-1))`
`= ((M^3+1)) / (M(M-1))`
`=` non-zero constant
`:. x^3+y^3 prop x^2y-xy^2`
2. If `x prop y,y prop z,` then prove that `x^3+y^3+z^3 prop xyz`
Solution:
`x prop y,y prop z`
`=>x=M*y` (where constant `M != 0`)
And `y=N*z` (where constant `N != 0`)
substituting `y=Nz` in `x=My`, We get `x=M(Nz)=MNz`
Now `(x^3+y^3+z^3) / (xyz)`
`= (M^3N^3z^3+N^3z^3+z^3) / (MN^2z^3)`
`= (z^3(M^3N^3+N^3+1)) / (MN^2z^3)`
`= ((M^3N^3+N^3+1)) / (MN^2)`
`=` non-zero constant
`:. x^3+y^3+z^3 prop xyz`
3. If `5x-7y prop 6x+3y,` then prove that `x prop y`
`5x-7y prop 6x+3y`
`=> 5x-7y = M(6x+3y)` (where constant `M != 0`)
`=> 5x-7y = 6Mx+3My`
`=> 5x-6Mx = 7y+3My`
`=> x(5-6M) = y(7+3M)`
`=> x/y = (7+3M) / (5-6M)`
`=> x/y = `non-zero constant
`=> x prop y`
4. If `x^2+9y^2 prop xy,` then prove that `x prop y`
Solution:
`x^2+9y^2 prop xy`
`=>x^2+9y^2=Mxy` (where constant `M != 0`)
`=> (x^2+9y^2) / (xy) = M`
`=> (x^2+9y^2) / (6xy) = M/6` (multiplying by `1/6`)
`=> (x^2+9y^2+6xy) / (x^2+9y^2-6xy) = (M+6)/(M-6)` (componendo and dividendo)
`=> (x+3y)^2 / (x-3y)^2 = (M+6)/(M-6)`
`=> (x+3y) / (x-3y) = sqrt((M+6)/(M-6))`
`=> (x+3y) / (x-3y) = N` (where `N=sqrt((M+6)/(M-6))` and `N!=0, M != +- 6, x != +- 3y`)
`=> (x+3y+x-3y) / (x+3y-x+3y) = (N+1)/(N-1)` (componendo and dividendo)
`=>2x / 6y = (N+1)/(N-1)`
`=>x / 3y = (N+1)/(N-1)`
`=>x / y=3((N+1)/(N-1))`
`=>x/y = `non-zero constant
`=>x prop y`
This material is intended as a summary. Use your textbook for detail explanation.
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