Problem 30 - Prealgebra

Question

Simplify expression. \(4 y-5 y\)

Step-by-Step Solution

Answer

The simplified expression is \(-y\).

Step 1: Identify Like Terms

The terms in the expression are "4y" and "-5y". Both of these terms are like terms because they both contain the variable "y" raised to the power 1.

Step 2: Subtract the Coefficients

Since the terms are like terms, you can combine them by subtracting their coefficients. The coefficient of the first term is 4, and the coefficient of the second term is -5. Subtract 5 from 4 to get the new coefficient: \[ 4 - 5 = -1 \]

Step 3: Write the Simplified Expression

Combine the result of the coefficients to write the simplified expression. Because the resulting coefficient is -1, the expression becomes:\[ -1y \] This can also be written simply as \[ -y \]

Key Concepts

Understanding Like Terms

In algebra, simplifying expressions often starts by identifying like terms. Like terms are terms in an expression that have the same variable and the same exponent. For example, if you see terms like \(4y\) and \(-5y\), they are considered like terms. This is because both terms contain \(y\) raised to the first power (essentially just \(y\)).

Recognizing like terms is crucial because it tells us which terms can be combined. Only terms that are classified as like terms can be operated upon directly. If two terms have different variables or exponents, they aren't like terms and should stay separate in the expression. Here are some quick reminders:

• Terms like \(3x^2\) and \(5x^2\) can be combined as they share both the variable \(x\) and the exponent \(2\).
• Terms such as \(2a\) and \(3b\) cannot be combined because they have different variables.
• Terms \(7m\) and \(4m^3\) cannot be combined because the exponents differ, even though the variable is the same.

Identifying like terms is a fundamental step in simplifying any algebraic expression. By doing this first, your path to simplification becomes much clearer.

The Role of Coefficients in Simplification

Once like terms are identified, the next step involves the coefficients. Coefficients are the numerical parts of the terms in an algebraic expression. For example, in the term \(4y\), "4" is the coefficient. Coefficients express how many times the variable is being multiplied.

When simplifying, we often need to combine coefficients of like terms. This is done through basic arithmetic operations such as addition or subtraction. In the example \(4y - 5y\):

• First, take note of the coefficients: they are 4 and -5.
• Combine them by performing the proper operation: subtract \(-5\) from \(4\), resulting in \(-1\).
• The term then becomes \(-1y\), which can also be written simply as \(-y\).

This shows that understanding and manipulating coefficients is key to efficiently simplifying algebraic expressions. Remember always to perform operations only on coefficients of like terms.

Introduction to Variable Manipulation

Variable manipulation is at the heart of simplifying algebraic expressions. In basic terms, it refers to the process of rearranging and combining terms in an algebraic expression. While it's closely tied to understanding both like terms and coefficients, variable manipulation ensures that you end up with the simplest form of the expression.

Let's explore this with \(4y - 5y\). Here, after identifying the like terms and their coefficients, you effectively manipulate the expression by simplifying it to \(-y\). This process involves

• Understanding the rules of arithmetic operations on coefficients.
• Knowing how the sign of a number affects the operations (e.g., subtracting a negative adds a positive).
• Rewriting the expression in its simplest form, which aids in understanding and solving further algebraic problems effectively.

The goal of variable manipulation is not only to simplify but also to prepare expressions for further operations, such as solving equations. Mastering variable manipulation is a powerful tool in your mathematical toolkit.