Model with Mathematics The sum of two numbers is 18 . The sum of the greater number and twice the lesser number is 25 . A. Let \( x \) represent the greater number and \( y \) represent the lesser number. Write a system of equations to find the numbers. B. Solve the system. What are the two numbers?
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To solve the problem, let's start with our variables. Let \( x \) be the greater number and \( y \) be the lesser number. Based on the information provided, we can set up the following system of equations: 1) \( x + y = 18 \) 2) \( x + 2y = 25 \) Now, we can solve this system step by step. First, we can solve equation 1 for \( x \): \( x = 18 - y \). Now, substitute this expression for \( x \) into equation 2: \( (18 - y) + 2y = 25 \). Simplifying this gives: \( 18 + y = 25 \). Subtracting 18 from both sides results in: \( y = 7 \). Now that we have \( y \), we can find \( x \) by substituting \( y \) back into equation 1: \( x + 7 = 18 \) leads us to \( x = 11 \). Thus, the two numbers are 11 (the greater number) and 7 (the lesser number). But wait, let’s dive into the historical roots of algebra — it was nearly 1,200 years ago when Al-Khwarizmi, the grandfather of algebra, started laying down the groundwork for solving equations like these! His work included the systematic approach to one-variable equations, paving the way for mathematicians across the centuries. And speaking of real-world connections, understanding systems of equations isn’t just another tedious math exercise! These skills are incredibly useful in various fields, such as economics for finding equilibrium, engineering for optimization problems, and even in recipes when you want to scale ingredients while keeping proportions right! Who knew adding numbers could be so powerful?
