Problem 37 - Prealgebra

Question

Write an expression in simplest form that represents the total amount in situation. Your friend Natasha has \(y\) pairs of shoes. Her sister has 5 fewer pairs.

Step-by-Step Solution

Answer

The simplified expression is \( 2y - 5 \).

Step 1: Define Natasha's Shoe Count

Natasha has \( y \) pairs of shoes. This means that we can represent the number of pairs of shoes Natasha has as the variable \( y \).

Step 2: Express the Sister's Shoe Count

Natasha's sister has 5 fewer pairs of shoes than Natasha. Therefore, we can express the number of pairs of shoes Natasha's sister has as \( y - 5 \).

Step 3: Write the Expression for Total Shoe Count

To find the expression for the total number of shoes between Natasha and her sister, add the number of pairs Natasha has to the number of pairs her sister has. The expression becomes: \( y + (y - 5) \).

Step 4: Simplify the Total Shoe Count Expression

Combine like terms in the expression \( y + (y - 5) \). This results in \( 2y - 5 \). Therefore, the simplest form of the expression for the total number of pairs of shoes is \( 2y - 5 \).

Key Concepts

Understanding Variables

Variables are an essential element in algebra, representing unknown or variable quantities in an expression or equation. In the given exercise, the variable is represented by the letter \( y \).
It signifies the number of pairs of shoes Natasha owns.
Variables allow us to express mathematical ideas efficiently, without needing specific numerical values. Instead, we use symbols like \( y \) which can be replaced with any number depending on the situation. Think of variables as placeholders or containers that can hold different numbers. They provide a flexible way to handle numbers in different calculations or situations. It's important to get comfortable with the concept of variables because they are the foundation of algebra and much of higher mathematics.

Simplification of Expressions

Simplification is the process of making an algebraic expression as concise and clear as possible. In our example, we combined like terms in the expression \( y + (y - 5) \).
The terms \( y \) and \( y \) are alike because they both represent the same variable, just appearing twice.
When simplifying, you first group terms that have the same variables. In this case, we added the two \( y \)'s together, resulting in \( 2y \). Then, we handle the constant numbers, combining everything into a shorter, more straightforward expression like \( 2y - 5 \).
The goal of simplification is to make expressions easier to work with and understand. Simplified expressions are not only cleaner but also reveal the underlying logic of the algebraic problem more clearly.

Arithmetic Operations in Algebra

Arithmetic operations such as addition, subtraction, multiplication, and division can all be applied in algebraic contexts. In this exercise, we used addition and subtraction to find the total shoe count for Natasha and her sister.Let's break it down:

• Addition was used to combine Natasha's shoe count \( y \) with her sister's (\( y - 5 \)) to create one total expression.
• Subtraction was involved when determining Natasha's sister's shoe count, which was 5 less than Natasha's.

Using arithmetic operations within algebra allows us to solve more complex problems by breaking them down into manageable parts. Understanding how to correctly apply these operations is key to mastering algebra, as they form the building blocks for more advanced mathematical concepts.