IIT JEE , AIEEE Forum - Mathematics , Algebra

chetan_kp

Find all integer solutions for x, y of the equation

( x + 2y - 1)³ + (2x + y - 2)³ + 27( 1 - x - y)³ = 18

nivedh_89

what do those '?' marks stand for.........???????????

nadeemoidu

First of all , note that

( x + 2y - 1) + (2x + y - 2) + 3 ( 1 - x - y) = 0

So you can use the formula

if ( a+ b + c ) =0 then a³ + b³ + c³ = 3abc

so

3 ( x + 2y - 1) (2x + y - 2) [ 3 ( 1 - x - y) ] = 18

=>
( x + 2y - 1) (2x + y - 2) ( 1 - x - y) = 2

Now since x and y are integers , each of the terms is equal to 1 , -1 , 2 or -2.

Again note that ( x + 2y - 1) + (2x + y - 2) = 3 ( 1 - x - y)

So ( x + 2y - 1) + (2x + y - 2) = -3

On solving we get x=0 , y=0 or x=1 , y=-1

I have skipped some steps in between , so try to do it completely using the above idea.

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