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. Define the variables: Let the two numbers be \(x\) and \(y\).
2. Write the system of equations based on the problem statement: The sum of the numbers is 17, so \(x + y = 17\), and the product of the numbers is 42, so \(x \times y = 42\).
3. Express one variable in terms of the other using the sum equation: From \(x + y = 17\), we get \(y = 17 - x\).
4. Substitute \(y = 17 - x\) into the product equation to form a quadratic equation: \(x \times (17 - x) = 42\) which simplifies to \$17x - x^2 = 42$.
5. Rewrite the quadratic equation in standard form: \(x^2 - 17x + 42 = 0\), then solve this quadratic equation using factoring, completing the square, or the quadratic formula to find the values of \(x\) and \(y\).

Key Concepts

System of Equations

A system of equations consists of two or more equations with the same set of variables. Solving the system means finding values for the variables that satisfy all equations simultaneously. In this problem, two variables represent the unknown numbers, and their sum and product form two equations to solve.

Formulating Equations from Word Problems

Translating a word problem into mathematical equations involves identifying relationships described in words. Here, the sum of two numbers equals 17, and their product equals 42, which can be expressed as x + y = 17 and xy = 42. This step is crucial for applying algebraic methods.

Solving Quadratic Equations

When one equation is expressed in terms of one variable, substituting into the other often leads to a quadratic equation. Solving this quadratic (by factoring, completing the square, or using the quadratic formula) yields the values of the variables. These solutions correspond to the numbers sought.