Solve each problem using a system of equations in two variables. See Example 6. Find two numbers whose ratio is 4 to 3 and are such that the sum of their squares is 100.

Verified step by step guidance

1. Let the two numbers be x and y.
2. According to the problem, the ratio of the two numbers is 4 to 3, so we can write the equation: \( \frac{x}{y} = \frac{4}{3} \).
3. From the ratio equation, express one variable in terms of the other: \( x = \frac{4}{3}y \).
4. The sum of the squares of the two numbers is 100, so we have the equation: \( x^2 + y^2 = 100 \).
5. Substitute the expression for x from the ratio equation into the sum of squares equation: \( \left(\frac{4}{3}y\right)^2 + y^2 = 100 \).

Key Concepts

Systems of Equations

A system of equations consists of two or more equations with the same set of variables. To solve such a system, one must find the values of the variables that satisfy all equations simultaneously. Common methods for solving systems include substitution, elimination, and graphing. Understanding how to manipulate and solve these equations is crucial for tackling 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 4 to 3, which can be expressed as the equation x/y = 4/3. This concept is essential for establishing a relationship between the two unknown numbers and allows for the formulation of one equation based on their proportionality.

Sum of Squares

The sum of squares refers to the total obtained by squaring each number in a set and then adding those squares together. In this context, the problem states that the sum of the squares of the two numbers equals 100, leading to the equation x² + y² = 100. This concept is important for creating a second equation that, when combined with the ratio equation, allows for the solution of the system.