Problem 36 - 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 \(25 .\) Two times the larger number is 12 times the smaller number. Find the numbers.

Step-by-Step Solution

Answer

The two numbers are 30 (the larger one) and 5 (the smaller one).

Step 1: Set up the system of equations

From the exercise, we have two conditions that allow us to generate two equations.\n1. The difference between two numbers is 25. Let's denote the larger number as \(x\) and the smaller one as \(y\). So, we get our first equation as \(x - y = 25\).\n2. Two times the larger number is 12 times the smaller number. This can be written as \(2x = 12y\). Now, we have a system of two equations: \(x - y = 25\) and \(2x = 12y\).

Step 2: Solve the system by substitution

Rewrite the second equation in terms of \(x\) as \(x = 6y\). Substitute \(x\) from this equation into the first equation: \(6y - y = 25\) or \(5y = 25\). Divide both sides by 5 to solve for \(y\), yielding \(y = 5\).

Step 3: Find the second number

Plug the value of \(y = 5\) into the equation \(x = 6y\) to find \(x\). This gives \(x = 6 * 5 = 30\).