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.

Verified step by step guidance

1. Define the variables: Let $x$ be the first number and $y$ be the second number.
2. Write the first equation based on the condition 'The difference between the squares of two numbers is 3': $x^2-y^2=3$.
3. Write the second equation based on the condition 'Twice the square of the first number increased by the square of the second number is 9': $2x^2+y^2=9$.
4. Solve the first equation for one of the variables, for example, express $y^2$ in terms of $x^2$: $y^2=x^2-3$.
5. Substitute $y^2=x^2-3$ into the second equation and solve for $x$: $2x^2+(x^2-3)=9$.

Key Concepts

Nonlinear Equations

Nonlinear equations are equations in which the variables are raised to a power greater than one or involve products of variables. Unlike linear equations, which graph as straight lines, nonlinear equations can produce curves, parabolas, or other complex shapes. Understanding how to identify and manipulate these equations is crucial for solving systems that involve relationships like the ones described in the problem.

System of Equations

A system of equations consists of two or more equations that share the same variables. The solution to the system is the set of values that satisfy all equations simultaneously. In this problem, we need to formulate a system based on the given conditions and then find the values of x and y that meet both criteria, which is essential for solving the problem.

Difference of Squares

The difference of squares is a specific algebraic identity that states a² - b² = (a - b)(a + b). This concept is useful in simplifying expressions and solving equations involving the squares of numbers. In the context of this problem, recognizing that the difference between the squares of two numbers can be expressed using this identity will aid in forming the correct equations for the system.