Problem 20 - Prealgebra

Question

Simplify the following expressions by combining similar terms. In some cases the order of the terms must be rearranged first by using the commutative property. $$4 y-3 y-7+2$$

Step-by-Step Solution

Answer

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

Step 1: Rearrange Terms

First, use the commutative property to rearrange the terms so that like terms are next to each other. The expression \(4y - 3y - 7 + 2\) can be rearranged to \((4y - 3y) + (-7 + 2)\).

Step 2: Combine Like Terms

Now, combine the like terms. Like terms have the same variable. Combine \(4y\) and \(-3y\) to get \(1y\) or just \(y\). The terms \(-7\) and \(+2\) add up to \(-5\).

Step 3: Final Simplification

After combining the terms, the simplified expression will be \(y - 5\).

Key Concepts

Commutative Property

The commutative property is a fundamental concept in prealgebra. It allows you to change the order of numbers in addition or multiplication without changing the result. This property is extremely handy when simplifying expressions. For example, in the expression \(4y - 3y - 7 + 2\), rearranging the terms makes it easier to combine like terms. The commutative property states that \(a + b = b + a\) and \(a \times b = b \times a\).

• Addition Example: \(3 + 5 = 5 + 3\)
• Multiplication Example: \(2 \times 4 = 4 \times 2\)

Using the commutative property helps group like terms next to each other, simplifying the process of solving or reducing expressions. It's a small but powerful tool in your algebra toolkit!

Simplifying Expressions

Simplifying expressions is the process of making a mathematical expression as simple as possible. This often involves combining like terms, which are terms that have the same variable raised to the same power.When simplifying, you want to make equations easier to work with by collecting all like terms together and making the expression shorter and more manageable. For instance, in the expression \((4y - 3y) + (-7 + 2)\), you combine the like terms:

• Like terms: Terms with the same variables. Here, \(4y\) and \(-3y\) are like terms.
• Constant terms: Just numbers, like \(-7\) and \(2\), are also like terms.

By combining \(4y\) and \(-3y\), you get \(1y\) or simply \(y\). For the numbers, \(-7 + 2\) simplifies to \(-5\). Thus, the expression simplifies neatly to \(y - 5\). The goal is to make sure your final expression is accurate and reflects all that you've calculated.

Prealgebra

Prealgebra is an introductory course to algebra that lays the foundation for high-level math courses. It covers basic arithmetic and moves on to more complex topics, like expressions and equations, much like simplifying expressions and using properties such as the commutative property.

Prealgebra focuses on:

• Understanding Variables: Recognize and work with variables, like \(y\), as placeholders in equations.
• Expressions: Learn how to simplify expressions using mathematical properties.
• Equations: Develop skills to manipulate and solve equations accurately.
• Mathematical Properties: Master important properties, such as commutative, associative, and distributive laws which are essential for solving equations.

Successful mastery of prealgebra concepts prepares students for more advanced algebra topics. It's like building a strong foundation for a house; crucial for subsequent stability.