Just manipulation

Algebra Level 2

{ e x + e y = 2 e x 3 + y 3 + 3 x y = 1 \large \begin{cases} e^{x} + e^{y} = 2\sqrt e \\ x^{3} +y^{3} + 3xy = 1 \end{cases}

Find the largest real value of x x satisfying the system of equations above.


The answer is 0.5.

Raghu Raman Ravi by Raghu Raman Ravi , 20, India

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1 solution

Raghu Raman Ravi
Sep 25, 2017

From the second equation x 3 + y 3 + ( − 1 ) 3 = 3 x y ( − 1 ) = > x + y + ( − 1 ) = 0 o r x = y = − 1 x^{3} + y^{3} + (-1)^{3} = 3xy(-1) => x+y+(-1) = 0 or x=y= -1 But x=y=-1 does not satisfy the first equation. Therefore, y = 1 − x y=1- x Substituting this in the first equation and simplifying, we get ( e x − e 1 / 2 ) 2 = 0 = > x = 1 / 2 (e^{x}-e^{1/2})^{2}= 0 => x= 1/2 Which is the only solution.

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