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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`.
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Solution
`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.
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