Problem 60 - Precalculus: Mathematics for Calculus

Question

Number Problem 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 Variables

Let's denote the smaller number by \( x \) and the larger number by \( y \). We need to express these numbers based on the problem's conditions.

Step 2: Set Up the First Condition

According to the problem, the sum of two numbers is twice their difference. This gives us the equation: \( x + y = 2(y - x) \). Simplifying this, we get \( x + y = 2y - 2x \).

Step 3: Simplify the First Equation

Rearrange the equation from Step 2: \( x + y = 2y - 2x \) to \( 3x = y \). This shows the relationship between the two numbers.

Step 4: Set Up the Second Condition

The larger number is 6 more than twice the smaller number, which gives us the equation: \( y = 2x + 6 \).

Step 5: Substitute and Solve Equations

Substitute \( y = 3x \) from Step 3 into the equation from Step 4, \( y = 2x + 6 \). This gives \( 3x = 2x + 6 \).

Step 6: Solve for \( x \)

From \( 3x = 2x + 6 \), solve for \( x \): Subtract \( 2x \) from both sides to get \( x = 6 \).

Step 7: Solve for \( y \)

Using the relationship \( y = 3x \), substitute \( x = 6 \) to find \( y \): \( y = 3 \times 6 = 18 \).

Key Concepts

Equation Solving

When solving algebraic word problems, we look to create equations from the given information. This process is called equation solving. It involves translating the words into mathematical expressions and equations. Here are the steps often involved in equation solving for word problems:

• Understand the problem: Carefully read the problem to determine what is given and what needs to be found.
• Translate the problem into equations: Use mathematical symbols to express the given information as one or more equations.
• Solve the equations: Use algebraic methods to find the values of the unknowns (variables).

For instance, in our exercise, we derived two equations from the given conditions to find our numbers. By manipulating these equations, we solved for the unknown values of the variables. Solving equations is a fundamental skill in algebra that allows you to find unknown quantities using known relationships.

Variable Definition

In algebra, defining variables is a crucial step in solving word problems. A variable is a symbol, often a letter like \( x \) or \( y \), that represents an unknown number or quantity. Here's how you can define variables effectively:

• Identify what quantities are unknown and need to be found.
• Choose symbols to represent these unknowns.
• Clearly state what each variable represents.

In our problem, we started by defining \( x \) as the smaller number and \( y \) as the larger number. This clear definition is important because it helps organize the information and make setting up equations easier. Once you have your variables defined, you can use them to express relationships and conditions given in the problem.

Linear Equations

Linear equations are equations of the first degree, meaning they have the highest power of the variable as one. They are important building blocks in algebraic problem-solving. A linear equation can be expressed in the form \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants.In our word problem, one of the derived equations was \( 3x = y \), which is a simple linear relationship. Linear equations often represent relationships between two variables and can be solved using substitution or elimination methods. In this problem, we substituted one equation into the other to find the value of one variable. Then, we used that value to solve for the second variable.Understanding how to work with linear equations is essential because they help us describe relationships in many real-world scenarios. By mastering the manipulation and solving of these equations, you can comfortably tackle a wide range of algebraic problems.