Solve This euqations and find tha value of x and y!!
Answers
Add the given equations. Refer the attachment
Use the respective identities and solve.
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Multiply by 3 on both sides of ( ii ),
Now, adding ( i ) and ( iii )
,
If we notice, ⁹√x is the cube root of ∛x and similarly ⁹√y is the cube root of ∛y and vice versa.
With the help of the identity a^3 + b^3 + 3ab( a + b ) = ( a + b )^3 of factorization, we can say that ∛x + ∛y + 3(⁹√xy)( ⁹√x + ⁹√y ) is equal to ( ⁹√x + ⁹√y )^3.
Cube root on both sides,
⇒
Substituting the value of in ( ii ),
Now, substitute the value of x in ( i ),
Let
⇒ 8 + k^2 = 9k
⇒ k^2 - 9k + 8 = 0
⇒ k^2 - ( 8 + 1 )k + 8 = 0
⇒ k^2 - 8k - k + 8 = 0
⇒ k( k - 8 ) - ( k - 8 ) = 0
⇒ ( k - 8 )( k - 1 ) = 0
∴ k = 8 or k = 1
⇒ k = 8 or k = 1
⇒ ∛y = 8 or ∛y = 1
⇒ y = 8^3 or y = 1^3
⇒ y = 512 or y = 1
From above, we calculated that xy = 2^9 = 512
⇒ xy = 512
⇒ x = 512 / 512 or x = 512 / 1
⇒ x = 1 or 512
Therefore the values of x and y are 512 and 1.
Please check as your answer is something different, shubhendu