In Exercises 43–46, let x represent one number and let y represent the other number. Use the given conditions to write a system of nonlinear equations. Solve the system and find the numbers. The difference between the squares of two numbers is 3. Twice the square of the first number increased by the square of the second number is 9. Find the numbers.

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Define the variables: let $x$ be the first number and $y$ be the second number.

Translate the first condition "The difference between the squares of two numbers is 3" into an equation: $x^{2} - y^{2} = 3$.

Translate the second condition "Twice the square of the first number increased by the square of the second number is 9" into an equation: $2x^{2} + y^{2} = 9$.

You now have the system of nonlinear equations: \[ \begin{cases} x^{2} - y^{2} = 3 \\ 2x^{2} + y^{2} = 9 \end{cases} \] Use substitution or elimination to solve this system. For example, add the two equations to eliminate $y^{2}$ or express $y^{2}$ from one equation and substitute into the other.

After finding $x^{2}$ and $y^{2}$, take the square roots to find possible values for $x$ and $y$. Remember to consider both positive and negative roots since squaring eliminates sign information.

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