Question

If `x/a = y/b = z/c`, prove that `(3x^3 - 5y^3 + 4z^3)/(3a^3 - 5b^3 + 4c^3) = ((3x - 5y + 4z)/(3a - 5b + 4c))^3`.

Answer 1

`x/a = y/b = z/c` = k(say)
x = ak, y = bk, z = ck

L.H.S. = `(3x^3  5y^3 + 4z^3)/(3a^3  5b^3 + 4c^3)`

= `(3a^3k^3 - 5b^3k^3 + 4c^3k^3)/(3a^3 - 5b^3 + 4ac^3)`

= `(k^3(3a^3 - 5b^3 + 4c^3))/(3a^3 - 5b^3 + 4c^3`
= k³
R.H.S. = `((3x - 5y + 4z)/(3a - 5b + 4c))^3`

= `((3ak - 5bk + 4ck)/(3a - 5b + ac))^3`

= `((k(3a - 5b + 4c))/(3a - 5b + 4c))^3`
= (k)³
= k³
∴ L.H.S. = R.H.S.