Question

if x andy be two positive real numbers such that 8x^3+27y^3=270 and 2x^y+3xy^2=15, then evaluate 2x+3y

Answer 1

$8x ^3 + 27y ^3 \\ 2x {} {}^{2} + 3xy { }^{2} \\ = (2x + 3y) { }^{2} = (2x) ^3 + (3y) ^3 + 3(2x)(3y)(2x + 3y) \\ = \\ = 8x ^3 + 27y ^3 + 18xy(2x + 3y) \\ =( 2 x + 3y) ^3 = 8x ^3 + 27y ^3 + 18(2x ^2 + 3xy ^2) \\ = (2x + 3y ) ^3 = 730 + 18(15) \\ = 730 + 270 =( 2x + 3y) ^3 = 1000 \\ =( 2x + 3y) ^3 = 1000 \\ = 2x + 3y = \sqrt[3]{1000 } \\ = 2x + 3y = 10 \\ \\ \\ thus \: required \: value \: of \: the \: expression \: is \: 10$