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Solve the system of nonlinear equations:

\[ \begin{align*} x^2 + y^2 &= 5 \\ xy &= 2 \end{align*} \]

1 Answer

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Final Answer:

The solution to the system of nonlinear equations is
\(x = √(2)\) and
\(y = √(3)\).

Step-by-step explanation:

To solve the system of equations
\(x^2 + y^2 = 5\) and \(xy = 2\), we can use substitution. First, express one of the variables in terms of the other using the second equation. Solving for \(y\), we get
\(y = (2)/(x)\). Substitute this expression for \(y\) into the first equation:


\[x^2 + \left((2)/(x)\right)^2 = 5.\]

Simplify this equation:


\[x^2 + (4)/(x^2) = 5.\]

Multiply through by
\(x^2\) to get rid of the fraction:


\[x^4 + 4 = 5x^2.\]

Rearrange to form a quadratic equation:


\[x^4 - 5x^2 + 4 = 0.\]

Factor this equation:


\[(x^2 - 1)(x^2 - 4) = 0.\]

This yields two possible values for \(x\): \(x = 1\) or \(x = -1\) from the first factor, and \(x = 2\) or \(x = -2\) from the second factor. However, considering the positive solutions, we have \(x = 1\) or
\(x = √(2)\).Substitute these back into the second equation \(xy = 2\) to find the corresponding values of \(y\). The solution
\((√(2), √(3))\) satisfies both equations. Therefore, the final answer is
\(x = √(2)\) and \(y = √(3)\).

User VilleKoo
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