Solve using the five "Steps for Solving Applied Problems." A...

Solve using the five "Steps for Solving Applied Problems."
A 55 -ft-long drainage pipe must be cut into two pieces before installation. One piece is two-thirds as long as the other. Find the length of each piece.

Key Concepts

Understanding the Problem

This concept involves carefully reading and comprehending the scenario presented by an applied problem. It requires identifying what is given, what is unknown, and the relationships between quantities. A clear understanding forms the foundation for a structured solution process by ensuring that all relevant information is considered before proceeding to mathematical modeling.

Variable Assignment

Variable assignment is the process of symbolically representing the unknown quantities in the problem with letters or symbols. This allows one to translate a verbal description of the scenario into an algebraic format. It is a critical step as it helps in organizing the information and setting the stage for forming equations that mirror the relationships described in the problem.

Formulating the Equation

This key concept involves translating the verbal relationships and constraints described in the problem into one or more algebraic equations. It requires an understanding of ratios, sums, differences, or other operations as dictated by the problem. A well-formulated equation accurately reflects the problem’s conditions and is essential for applying algebraic methods to reach a solution.

Solving Linear Equations

After formulating the necessary equation, the next step is to solve it using algebraic techniques. Solving linear equations involves combining like terms, isolating the variable, and performing arithmetic operations to find the unknown value. This step is fundamental in algebra as it converts the abstracted problem into a concrete numerical solution.

Solution Verification

Verification is the final key concept where the obtained solution is checked against the original problem conditions to ensure accuracy. This step involves substituting the solution back into the initial equations or problem context to confirm that all conditions are satisfied. It is a crucial practice in problem-solving that helps catch any errors and validates the correctness of the result.

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