Consider the equation #log_10 |x^3 +y^3| - log_10 |x^2-xy +y^2| +log_10 |x^3-y^3| - log _10 |x^2+xy+y^2| = log_10(221)#. If #x+y=17#, then the sum of all possible values of #x# is?

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Answer 1 · 2018-07-14T11:25:55

# 15#.

We have, #log_10(x^3+y^3)-log_10(x^2-xy+y^2)#,

#=log_10{(x^3+y^3)/(x^2-xy+y^2)}#,

#=log_10{((x+y)(x^2-xy+y^2))/(x^2-xy+y^2)}#,

#=log_10(x+y)#.

Similarly, #log_10(x^3-y^3)-log_10(x^2+xy+y^2)#

#=log_10(x-y)#.

Utilising these, the given eqn. becomes,

#log_10(x+y)+log_10(x-y)=log_10 221#.

#:. log_10{(x+y)(x-y)}=log_10 221#.

#:. (x+y)(x-y)=221............(1)#.

But, we are given that, #x+y=17....(2)#.

Then, by #(1), x-y=221/17, i.e., x-y=13....(3)#.

Solving #(2) and (3), x=15#.

Thus, the only possible values of #x# for which the eqn. holds is

#x=15#.

Hence, the desired sum is #15#.