Problem 38 - Introductory Algebra for College Students

Question

Write a system of equations modeling the given conditions. Then solve the system by the substitution method and find the two numbers. The difference between two numbers is \(5 .\) The sum of the larger number and twice the smaller number is \(14 .\) Find the numbers.

Step-by-Step Solution

Answer

The larger number is 8 and the smaller number is 3.

Step 1: Formulate the System of Equations

Let the two numbers be \(x\) and \(y\). Formulate the first equation from the condition that the difference between the two numbers is \(5\), which can be written as: \(x - y = 5\). The second equation is the sum of the larger number and twice the smaller number equals \(14\). As \(x\) is larger, it can be written as: \(x + 2y = 14\).

Step 2: Use Substitution Method to Solve the System

Now solve the first equation for \(x\), that results in: \(x = y + 5\). Now you can substitute \(x\) in the second equation with \(y + 5\): \(y + 5 + 2y = 14\). Now simplify the equation: \(3y + 5 = 14\). Lastly, solve the equation for \(y\) which results in: \(3y = 9\) and \(y = 3\). Now substitute \(y = 3\) into either of the original equations to solve for \(x\). Using the first equation \(x - y = 5\) gives: \(x - 3 = 5\) and \(x = 8\).

Step 3: Verify your Solution

Substitute the values \(x = 8\) and \(y = 3\) back into the original system to verify that they are the solution to the system. \(8 + 2 * 3 = 14\) and \(8 - 3 = 5\). Both these equations are correct, so the solution to the system is \(x = 8\) and \(y = 3\).