Problem 58 - Precalculus Mathematics for Calculus

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 call the smaller number \( x \) and the larger number \( y \). We are given two relationships between them.

Step 2: Translate the First Condition

The first condition states that the sum of the two numbers is twice their difference. This can be written as: \[ x + y = 2(y - x) \]

Step 3: Simplify the Equation

Expand and simplify the equation from Step 2:\[ x + y = 2y - 2x \] Rearrange terms to get all terms involving the same variable on one side:\[ 3x = y \]

Step 4: Translate the Second Condition

The second condition states that the larger number is 6 more than twice the smaller. This gives us another equation:\[ y = 2x + 6 \]

Step 5: Substitute and Solve for the Smaller Number

Substitute the expression for \( y \) from Step 3 into the equation obtained from Step 4:\[ 3x = 2x + 6 \]Solve for \( x \):\[ x = 6 \]

Step 6: Find the Larger Number

Use the value of \( x \) to find \( y \). Substitute \( x = 6 \) into \( y = 3x \):\[ y = 3(6) = 18 \]

Step 7: Verify the Solution

Verify that these values satisfy both original conditions:1) Check: Sum is twice the difference: \[ 6 + 18 = 24 \] and \[ 2(18 - 6) = 24 \].2) Check: Larger is 6 more than twice the smaller: \[ 18 = 2 \times 6 + 6 = 18 \].Both conditions are satisfied.

Key Concepts

Equation Solving

Solving equations is a fundamental skill in algebra.
It involves finding the value of unknowns, typically represented by variables, that make an equation true. In equation solving, you work with the principles of balance: whatever you do to one side of the equation, you must do to the other. There are key steps in solving equations, such as:

• Identifying the equation and writing it down clearly.
• Simplifying the equation by combining like terms or using distributive property.
• Isolating the variable on one side of the equation to find its value.
• Checking your solution by substituting back into the original equation.

Equation solving requires practice and attention to detail. Understanding each step helps build confidence in handling more complex problems.

Variables

Variables are symbols used in mathematics to represent numbers or quantities that may vary.
In algebra, the most common symbols used as variables are letters such as \( x \), \( y \), or \( z \). They allow us to write general rules and explore mathematical relationships.Here’s why variables are valuable:

• They help to generalize mathematical statements and solve problems with unknown values.
• Variables make it possible to describe functions and formulas where values are not constant.
• They facilitate the formulation of equations which can represent real-world scenarios.
• Using variables allows for easy substitution once values are known.

Working with variables is central to algebra. They provide a flexible way to solve problems and find solutions that are applicable in varied situations.

Linear Equations

Linear equations are algebraic equations in which each term is either a constant or the product of a constant and a single variable.
They are called linear because they graph as straight lines on a coordinate plane.Linear equations often take the form \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants.
Solving linear equations involves finding the value of the variable that makes the equation true.Here are some characteristics of linear equations:

• They have one or two variables, but each variable is raised only to the power of one.
• Simplifying and analyzing these equations involves basic arithmetic manipulations.
• When graphed, they produce a straight line, hence the name "linear".
• Linear equations model many real-world situations, such as predicting trends and solving problems involving rates.

Understanding linear equations is crucial for solving a range of algebraic problems. They form the foundation for more advanced mathematical concepts.