Question

Solve the system of nonlinear equations:

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

Answer 1

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 \(y = √(3)\).`