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 \(y\). According to the problem, their ratio is 4 to 3, which can be written as the equation \(\frac{x}{y} = \frac{4}{3}\).
2. Rewrite the ratio equation as \$3x = 4y\( or equivalently \)x = \frac{4}{3}y$ to express one variable in terms of the other.
3. The problem also states that the sum of their squares is 100, which gives the equation \(x^2 + y^2 = 100\).
4. Substitute the expression for \(x\) from the ratio equation into the sum of squares equation: \(\left(\frac{4}{3}y\right)^2 + y^2 = 100\).
5. Simplify the equation and solve for \(y\). Once you find \(y\), use \(x = \frac{4}{3}y\) to find the value of \(x\).

Key Concepts

Systems 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 equations will represent the ratio and the sum of squares conditions.

Ratio and Proportions

A ratio compares two quantities, showing how many times one value contains or is contained within the other. Here, the ratio 4 to 3 means the two numbers can be expressed as 4x and 3x, which helps form one equation relating the variables.

Sum of Squares

The sum of squares refers to adding the squares of two numbers. In this problem, the sum of the squares of the two numbers equals 100, which forms the second equation in the system and helps determine the specific values of the numbers.