Problem 27 - Algebra 1: Concepts and Skills

Question

In Exercises 27–29, match the situation with the corresponding linear system. You have 7 packages of paper towels. Some packages have 3 rolls, but some have only 1 roll. There are 19 rolls altogether.

Step-by-Step Solution

Answer

The number of packages that contain 3 rolls is 6 and the number of packages that contain 1 roll is 1.

Step 1: Define the Variables

Let \(x\) be the number of packages that contain 3 rolls and \(y\) be the number of packages that contain 1 roll.

Step 2: Write the Equations from the Problem

From the information given, two equations can be written: Total number of packages: \(x + y = 7\) and total number of rolls: \(3x + y = 19\). Thus the corresponding linear system to represent the situation becomes: \[\begin{cases} x + y = 7\\ 3x + y = 19 \end{cases}\]

Step 3: Solve the System

To solve this system, subtract the first equation from the second to get \(2x = 12\) which simplifies to \(x = 6\). Substituting \(x = 6\) into the first equation gives \(y = 7 - 6 = 1\).

Key Concepts

Systems of Equations

Systems of equations are a set of two or more equations that we deal with together because they share variables. In the problem we discussed, we have a linear system composed of two equations:

• The first equation, \( x + y = 7 \), represents the total number of packages.
• The second equation, \( 3x + y = 19 \), accounts for the total number of rolls in those packages.

Each equation in the system provides a different piece of the puzzle. By using both equations, we can find the unique values of the variables \( x \) and \( y \). Solving these systems can show the best approach for packaging, planning, and even budgeting in real-life situations.
These equations also demonstrate a fun aspect of algebra, where we translate words into numbers, revealing how many possible solutions a situation might have. Often, systems of equations are drawn as lines on a graph, and solving them means finding where these lines intersect, representing the solution for both equations at once.

Variables in Algebra

Variables serve as placeholders for unknown numbers. They enable us to describe mathematical relationships succinctly. In the exercise, we defined:\

• \( x \) as the number of packages containing 3 rolls.
• \( y \) as the number of packages containing 1 roll.

By defining variables, we transform a word problem into a more manageable format with mathematical equations.
Variables also give us the flexibility to manipulate equations and explore the problem in different ways. As we solved for \( x \) and \( y \), these variables became concrete numbers, showing how many packages per type exist based on our conditions.
Using variables is fundamental in algebra. It allows us to understand complex relations and predict outcomes in various scenarios, from simple calculations to more intricate mathematical models. Variables help in making logical reasonings, step by step, become very understandable.

Solving Equations Step by Step

The key to successfully solving systems of equations is a clear, step-by-step process. Here's a breakdown:

• Step 1: Define the variables. This turns the word problem into a mathematical one.
• Step 2: Formulate the equations. We created two equations: \( x + y = 7 \) and \( 3x + y = 19 \). Each relates the different parts of the problem by using our variables.
• Step 3: Solve the system. For example, subtract the first equation from the second to simplify the system. This left us with \( 2x = 12 \), allowing us to solve for \( x = 6 \).
• Step 4: Substitute and find the other variable. By finding \( x \), substitute back into one of the original equations to discover \( y = 1 \).

Moving through these steps makes the solution clearer and manageable. Each step builds on the previous one, allowing you to gradually uncover the entire picture. This structure isn’t just for math problems; it's also a good way to approach lots of everyday decisions logically and effectively. The satisfaction of finding a solution can be quite rewarding, aiding the development of problem-solving skills applicable in various disciplines.