Solve each problem using a system of equations in two variables. See Example 6. Find two numbers whose ratio is 9 to 2 and whose product is 162.

Verified step by step guidance

1. Let the two numbers be represented by variables $x$ and $y$.
2. According to the problem, the ratio of the two numbers is 9 to 2. This can be expressed as the equation $\frac{x}{y}=\frac{9}{2}$.
3. Cross-multiply to eliminate the fraction: $2x=9y$.
4. The problem also states that the product of the two numbers is 162, which gives us the equation $xy=162$.
5. Now, solve the system of equations: $2x=9y$ and $xy=162$ to find the values of $x$ and $y$.

Key Concepts

Systems of Equations

A system of equations consists of two or more equations with the same variables. To solve such a system, one must find the values of the variables that satisfy all equations simultaneously. Common methods for solving include substitution, elimination, and graphing. Understanding how to manipulate and solve these equations is crucial for finding the desired values in problems involving relationships between variables.

Ratios

A ratio is a relationship between two quantities, indicating how many times one value contains or is contained within the other. In this problem, the ratio of two numbers is given as 9 to 2, which can be expressed as the fraction 9/2. This means that for every 9 parts of the first number, there are 2 parts of the second number. Understanding ratios is essential for setting up equations that reflect the relationships between the numbers.

Products of Numbers

The product of two numbers is the result of multiplying them together. In this context, the problem states that the product of the two unknown numbers is 162. This relationship can be expressed as an equation, which, along with the ratio, forms a system of equations. Recognizing how to formulate and solve equations based on products is key to finding the specific values of the numbers in question.