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 \(x\) and \(y\). According to the problem, their ratio is 9 to 2, which can be written as the equation \(\frac{x}{y} = \frac{9}{2}\).
2. Rewrite the ratio equation to express one variable in terms of the other. For example, multiply both sides by \(y\) to get \(x = \frac{9}{2} y\).
3. The problem also states that the product of the two numbers is 162, which gives the equation \(x \times y = 162\).
4. Substitute the expression for \(x\) from the ratio equation into the product equation: \(\left( \frac{9}{2} y \right) \times y = 162\).
5. Simplify the equation to get a quadratic in terms of \(y\): \(\frac{9}{2} y^2 = 162\). Then solve for \(y\), and use the value of \(y\) to find \(x\) using \(x = \frac{9}{2} y\).

Key Concepts

Systems of Equations

A system of equations consists of two or more equations with the same variables. Solving the system means finding values for the variables that satisfy all equations simultaneously. In this problem, two equations will represent the ratio and product conditions for the two numbers.

Ratio and Proportion

A ratio compares two quantities, showing how many times one value contains or is contained within the other. Expressing the ratio 9 to 2 means the two numbers can be represented as 9x and 2x, where x is a common multiplier. This helps form one equation in the system.

Product of Two Numbers

The product of two numbers is the result of multiplying them together. Given the product is 162, this forms the second equation in the system: (first number) × (second number) = 162. Using the ratio expressions, this equation helps solve for the unknown variable.