Problem 44 - Algebra and Trigonometry

Question

The sum of two numbers is twice their difference. The larger number is 6 more than twice the smaller. Find the numbers.

Step-by-Step Solution

Answer

The numbers are 6 and 18.

Step 1: Define the Variables

Let's define the two numbers. Let the smaller number be \( x \) and the larger number be \( y \).

Step 2: Set Up Equations Based on the Problem Statement

According to the problem, "The sum of two numbers is twice their difference." We can write this as \( x + y = 2(y - x) \). Also, "The larger number is 6 more than twice the smaller," which gives us the second equation: \( y = 2x + 6 \).

Step 3: Simplify the First Equation

Rearrange and simplify the equation from Step 2: \( x + y = 2(y - x) \). This simplifies to \( x + y = 2y - 2x \). Combine like terms to get \( 3x = y \).

Step 4: Substitute and Solve

Substitute \( y = 3x \) from Step 3 into the second equation \( y = 2x + 6 \). We get \( 3x = 2x + 6 \). Solve for \( x \): \( x = 6 \).

Step 5: Find the Larger Number

Now that we know \( x = 6 \), substitute it back into \( y = 3x \) to find \( y \). Thus, \( y = 3(6) = 18 \).

Step 6: Verify the Solution

Check if both original conditions hold: The sum \( x + y = 6 + 18 = 24 \) is twice their difference \( 2(y-x) = 2(18-6) = 24 \), and the larger number \( y = 18 \) is 6 more than twice the smaller: \( 2x + 6 = 2 \times 6 + 6 = 18 \). Both conditions are satisfied.

Key Concepts

Understanding Systems of Equations

A system of equations is a set of two or more equations with the same variables. In algebra, solving a system helps us find the values of these variables. For this problem, we have two equations involving our two unknowns, the smaller number \(x\) and the larger number \(y\).

• The first equation is based on the problem statement: "The sum of two numbers is twice their difference," which becomes \(x + y = 2(y - x)\).
• The second equation from the statement "The larger number is 6 more than twice the smaller," translates to \(y = 2x + 6\).

We need both equations because they give us two pieces of information about the relationship between \(x\) and \(y\). Solving systems of equations often provides a full solution that meets all given conditions. This is a fundamental aspect of algebraic problem-solving and helps us find precise answers that satisfy all constraints.

Utilizing Variable Substitution

Variable substitution is a technique where one variable is replaced with an expression containing another variable. In this exercise, after simplifying the first equation, we found a key relationship: \(3x = y\). This allowed us to substitute for \(y\) in the second equation. Substitution makes solving equations simpler by reducing the number of variables.

• We took \(y = 3x\) from our simplification of the first equation.
• Then, we substituted \(y = 3x\) into the second equation \(y = 2x + 6\), creating a single-variable equation: \(3x = 2x + 6\).

By solving this, we easily found \(x = 6\). With \(x\) known, we could substitute back to find \(y = 18\). Variable substitution is powerful because it can make complex problems much more manageable.

Problem Verification Process

Verification in algebra means checking our solution meets the original conditions or equations. This step is crucial to ensure accuracy and correctness of the solution. For our problem, after calculating \(x = 6\) and \(y = 18\), we verified by plugging these values back into the original statements:

• The sum condition: \(x + y = 6 + 18 = 24\), and twice the difference condition: \(2(y - x) = 2(18 - 6) = 24\).
• The larger number condition: \(y = 18\), and comparing with: \(2x + 6 = 2\times6 + 6 = 18\).

Both statements confirmed our solution, meaning the values satisfy the given equations. Verification is not just checking for mistakes; it's a logical step ensuring the solution integrity. Always double-check through original conditions to reinforce confidence in your solution.

Exploring Mathematical Reasoning

Mathematical reasoning in problem-solving involves logical thinking to understand, interpret, and solve problems effectively. In this exercise:

• We started by defining variables \(x\) and \(y\), an essential step to translate words into mathematical statements.
• Next, setting up equations based on problem descriptions involved critical reasoning to transform phrases into mathematical expressions.
• Simplifying an equation requires recognizing equivalent forms and ensuring balanced operations, showcasing reasoning skills to progress systematically.

Solving equations via substitution and verifying results highlight different reasoning aspects. Thoughtful examination and methodical practice hone these skills, ensuring solutions make sense and align with initial problem statements. Mathematical reasoning empowers us to unravel complex problems uniquely, using logic and structured approaches to reach reliable conclusions.