Problem 103 - Intermediate Algebra

Question

Perform each indicated operation. See Sections 1.4 and 5.4 $$ 9 y^{2}-8 y^{2} $$

Step-by-Step Solution

Answer

The result is \(y^2\).

Step 1: Identify Like Terms

First, identify the terms that are like terms in the expression. In this case, both terms are like terms because they contain the same variable part, which is \(y^2\).

Step 2: Perform the Subtraction Operation

Since both terms are like terms, subtract the coefficient of the second term from the coefficient of the first term. This operation is: \(9y^2 - 8y^2\). Perform this subtraction: \(9 - 8 = 1\).

Step 3: Rewrite the Expression

After performing the subtraction, replace the original expression with the resulting term, \(1y^2\), which simplifies to \(y^2\).

Key Concepts

Basic Algebra

In basic algebra, we often encounter problems that require us to manipulate and simplify mathematical expressions. Understanding the components of these expressions, such as terms and coefficients, is essential. A term in algebra is a product of numbers and variables raised to certain powers. For example, in the term \(9y^2\), '9' is called the coefficient, and \(y^2\) is the variable part. By identifying and understanding these elements, we lay the groundwork for simplifying algebraic expressions.

Learning basic operations, such as addition and subtraction of terms, is key. These operations are the foundation that allows us to solve equations and expressions involving variables. Remember, in basic algebra, you only combine terms that are alike, meaning they have the same variable raised to the same power.

Polynomial Subtraction

Polynomial subtraction involves subtracting the terms of two polynomials. It's similar to polynomial addition but involves taking away the coefficients of like terms.

Here's how polynomial subtraction works:

• Identify like terms - Terms that have the same variable parts.
• Subtract the coefficients of these like terms.
• Simplify the expression if needed.

In our example, we had to subtract two like terms \(9y^2\) and \(8y^2\). Since both terms contain \(y^2\), we simply subtract their coefficients: \(9 - 8 = 1\). This leaves us with \(1y^2\), simplifying to \(y^2\) in the end.

Simplifying Expressions

Simplifying expressions is a key skill in algebra. It involves reducing an expression to its simplest form. The goal is to make the expression easier to work with, often by combining like terms or performing arithmetic operations.

Here are steps to simplify:

• Identify and combine like terms - These are terms that have the same variable parts.
• Perform arithmetic operations like addition or subtraction on coefficients.
• Rewrite the expression in its simplest form.

In our exercise, once we performed the subtraction of the coefficients (\(9 - 8 = 1\)), we rewrote the expression. The term \(1y^2\) is simplified to \(y^2\) since the coefficient of '1' is usually omitted in algebraic expressions.

Remember, accurate simplification makes further algebraic manipulations much more manageable.