Find all solutions of the equation.
\[ y = -x \]
\[ y = x \ln x \]
Answer (2)
\left\{\begin{array}{ccc}y=-x\\y=xlnx\end{array}\right;\ \ \ D:x\in\mathbb{R^+}\\\\xlnx=-x\\\\xlnx+x=0\\\\x(lnx+1)=0\iff x=0\notin D\ or\ lnx+1=0\\\\lnx=-1\iff x=e^{-1}\to x=\frac{1}{e}\in D\\\\Answer:x=\frac{1}{e}\ and\ y=-\frac{1}{e}
The only solution for the equations y = − x and y = x ln x is ( x , y ) = ( e 1 , − e 1 ) . This was found by setting the equations equal to each other and solving for x . After finding x , we determined y using one of the original equations.
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