Problem 55 - College Algebra and Calculus: An Applied Approach

Question

A rectangular classroom seats 72 students. If the seats were rearranged with three more seats in each row, the classroom would have two fewer rows. Find the original number of seats in each row.

Step-by-Step Solution

Answer

The original number of seats in each row was 8.

Step 1: Define the Variables

Let x represent the original number of rows, and y represent the original number of seats in each row. Then the product xy gives the total number of seats.

Step 2: Formulate the First Equation

Since there are 72 seats, form the first equation as xy = 72.

Step 3: Formulate the Second Equation

If three more seats are put in each row, which increases y by 3, the classroom would only need x-2 rows to fit the same number of students. So, form the second equation as (y + 3)(x - 2) = 72.

Step 4: Solve the System of Equations

Now we have a system of two equations: xy = 72 and (y + 3)(x - 2) = 72. We can substitute the first one into the second one to solve for one variable, then substitute back to solve for the other variable.

Step 5: Find the original number of seats in each row

Solving the system of equations gives y = 8, which means originally, there were 8 seats in each row.

Key Concepts

Algebraic Problem-Solving

Algebraic problem-solving involves finding unknown variables that make a certain statement or series of statements true. It's the process of reasoning that is used to solve algebraic expressions and equations. In educational contexts, it's crucial for students to understand not just how to manipulate symbols, but also the reasoning behind the steps they take.

Considering the classroom seating problem, we began by defining variables to represent the unknowns: the original number of rows (\(x\)) and the original number of seats in each row (\(y\)). Defining variables is a foundational step in solving algebraic problems, as it allows us to translate word problems into mathematical expressions and equations. Through the step-by-step solution detailed above, we can see how algebraic manipulation, substitution, and logical reasoning were utilized to arrive at the final solution. This exercise is a prime example of algebraic problem-solving, turning a real-world scenario into an algebraic puzzle.

Systems of Linear Equations

A system of linear equations consists of two or more equations with two or more variables that are meant to be solved simultaneously. In the context of the classroom problem, we are given a system with two unknowns, which we represent with \(x\) and \(y\). The system is comprised of two equations: \(xy = 72\) and \( (y + 3)(x - 2) = 72\).

By representing a real-world problem with multiple relational expressions that share the same variables, these systems can be solved using one of several methods; including graphing, substitution, elimination, or matrix operations. The substitution method, as shown in our solution, is very powerful and often used when one equation can be easily isolated to express one variable in terms of another. Finding the intersection of these expressions gives us the one solution that satisfies both equations, thus solving our original problem.

Mathematical Modeling

Mathematical modeling is a method used to describe and analyze real-world situations in mathematical terms. It involves creating mathematical representations of a scenario that can then be manipulated and studied to predict and understand complex systems. This process often involves making assumptions and approximations to simplify the problem.

In our classroom seating example, we've used mathematical modeling to create a system of equations representing the physical layout and constraints of the seating arrangement. This type of model bridges the gap between a real-life scenario and an algebraic framework, allowing us to use mathematical tools to explore and solve real-world problems. This practical application illustrates how mathematical models can turn theoretical mathematics into a powerful tool for problem-solving in a variety of contexts.