Problem 63 - Beginning and Intermediate Algebra

Question

Simplify each expression. \(6 y^{2}+11 y^{2}-8 y^{2}\)

Step-by-Step Solution

Answer

The simplified expression is \ 9y^{2} \.

Step 1: Combine Like Terms

First, identify like terms in the expression. Here, all terms are like terms because they all have the same variable part, which is \(y^{2}\).

Step 2: Add and Subtract Coefficients

Add and subtract the coefficients of the like terms: \(6 + 11 - 8\). This gives \(9\).

Step 3: Write the Simplified Expression

Attach the simplified coefficient to the common variable part \(y^{2}\). The simplified expression is \ 9y^{2} \.

Key Concepts

What are Like Terms?

In algebra, like terms are terms that have the same variable part with the same exponent. This means they can be combined through addition or subtraction. In the expression we are working with,

• 6y², 11y², and -8y²

The variable part for all three terms is y², hence they are like terms. By identifying like terms, you only need to focus on their coefficients when simplifying. Remember, like terms must have exactly the same variables raised to the same powers. Numbers without variables are also considered like terms.

Understanding Coefficients

Coefficients are the numerical parts of terms in an algebraic expression. In the expression 6y², 11y², and -8y²:

• 6 is the coefficient of y² in the first term.
• 11 is the coefficient of y² in the second term.
• -8 is the coefficient of y² in the third term.

When simplifying expressions involving like terms, you'll be adding or subtracting these coefficients. For our example, we combined the coefficients 6, 11, and -8 to get 9. The variable part (y²) remains the same.

Creating a Simplified Expression

A simplified expression is one where all like terms have been combined, and there are no parentheses left. In our example, we started with 6y² + 11y² - 8y².

• First, we identified our like terms.
• Then, we added their coefficients together: 6 + 11 - 8.

This gave us a new coefficient of 9. Finally, we attached this new coefficient to the variable part y². So, the simplified expression is 9y². Simplifying expressions makes them easier to work with, especially in more complex algebraic operations.