Solve each problem using a system of equations in two variables. See Example 6. Find two numbers whose sum is 17 and whose product is 42.

Verified step by step guidance

1. Step 1: Define the variables. Let the two numbers be x and y.
2. Step 2: Set up the equations based on the given conditions. The first equation will be based on the sum: x + y = 17. The second equation will be based on the product: xy = 42.
3. Step 3: Solve one of the equations for one variable. For example, from the first equation, solve for y: y = 17 - x.
4. Step 4: Substitute the expression for y from Step 3 into the second equation. This will give you an equation in terms of x only: x(17 - x) = 42.
5. Step 5: Solve the quadratic equation obtained in Step 4. Expand and rearrange the terms to form a standard quadratic equation, then use the quadratic formula or factorization method to find the values of x.

Key Concepts

System of Equations

A system of equations is a set of two or more equations with the same variables. In this context, we will use two equations to represent the relationships between the two unknown numbers. Solving the system allows us to find values for the variables that satisfy all equations simultaneously.

Linear Equations

Linear equations are equations of the first degree, meaning they graph as straight lines. In this problem, we can express the conditions of the sum and product of the two numbers as linear equations. Understanding how to manipulate and solve these equations is crucial for finding the unknowns.

Factoring Quadratics

Factoring quadratics involves rewriting a quadratic equation as a product of its linear factors. In this problem, after setting up the equations, we may derive a quadratic equation from the product condition. Factoring allows us to find the roots, which correspond to the values of the two numbers we are trying to determine.