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 sum of two numbers is 10 and their product is 24. Find the numbers.

Verified step by step guidance

1. Define the variables: let \(x\) represent one number and \(y\) represent the other number.
2. Write the first equation based on the sum condition: \(x + y = 10\).
3. Write the second equation based on the product condition: \(x \times y = 24\).
4. Use the first equation to express one variable in terms of the other, for example, \(y = 10 - x\).
5. Substitute \(y = 10 - x\) into the second equation to get a quadratic equation: \(x(10 - x) = 24\), which simplifies to \$10x - x^2 = 24\(. Rearrange it to standard form: \)x^2 - 10x + 24 = 0\(. Then solve this quadratic equation using factoring, completing the square, or the quadratic formula to find the values of \)x\(, and subsequently \)y$.

Key Concepts

System of Nonlinear Equations

A system of nonlinear equations consists of two or more equations involving variables where at least one equation is nonlinear. In this problem, the sum and product conditions form equations that must be solved simultaneously to find the values of x and y.

Sum and Product of Two Numbers

The sum of two numbers x and y is expressed as x + y, and their product as x * y. These relationships help form equations based on given conditions, which are essential for setting up the system to solve for the unknown numbers.

Solving Quadratic Equations

After expressing one variable in terms of the other using the sum equation, substituting into the product equation leads to a quadratic equation. Solving this quadratic using factoring, completing the square, or the quadratic formula yields the possible values of the numbers.