Problem 51 - Algebra 1

Question

SIMPLIFYING EXPRESSIONS Simplify the expression by combining like terms. $$ -8 b-9 b $$

Step-by-Step Solution

Answer

The simplified expression is \(-17b\).

Step 1: Identify Like Terms

Looking at the expression \(-8b - 9b\), the like terms are \(-8b\) and \(-9b\). Both terms have \(b\) as their variable.

Step 2: Combine Like Terms

Like terms can be combined by adding or subtracting their coefficients. Here, we subtract the coefficient of \(-9b\) from the coefficient of \(-8b\). This gives us \(-8 -9\).

Step 3: Perform operation

Performing the operation \(-8 -9\) will yield the result of \(-17\).

Step 4: Write combined term

After performing the operation, the resulting coefficient needs to be written along with the common variable. The resulted term is \(-17b\).

Key Concepts

Combining Like Terms

Combining like terms is a fundamental skill in algebra. It helps to simplify expressions by collecting all similar elements together. In this process, we look for terms that have the same variable raised to the same power. These terms can then be merged into one term by adding or subtracting their coefficients. Let's dive deeper:

• To begin, identify terms that have the identical variable. For instance, in the expression \(-8b - 9b\), both terms include the variable \(b\).
• After identifying the like terms, focus on their numerical coefficients, the numbers in front of the variables, which are \(-8\) and \(-9\) here.
• Combine these coefficients by performing the arithmetic operation, which is typically addition or subtraction, depending on the sign of each term.

After combining, you write the new coefficient with the shared variable. For our example, the combined expression results in \(-17b\), simplifying the original expression to a single term.

Coefficients

Coefficients are the numbers placed directly in front of variables in an algebraic expression. They determine how many times the corresponding variable is being counted. They are crucial when combining like terms because they are the parts of the terms we actually combine.

• In the expression \(-8b - 9b\), \-8\ and \-9\ are the coefficients of \(b\).
• When combining like terms, you perform operations on these coefficients to simplify the expression.

Working with Coefficients

Consider the expression \(-8b - 9b\):

• We focus only on \(-8\) and \(-9\).
• Perform the operation \(-8 - 9\) to get \(-17\).
• The result is a single term with the coefficient \(-17\) and the variable \(b\).

Understanding and manipulating coefficients correctly is essential for simplifying expressions efficiently and accurately.

Like Terms

Like terms are terms in an algebraic expression that contain the same variables raised to the same exponents. Recognizing these terms is the first step in simplifying expressions.

• In any expression, you should look for terms that have identical variables and powers, such as \(b\) in \(-8b\) and \(-9b\).
• These terms can be combined because they represent the same quantity being counted multiple times, just in different amounts.

How to Identify Like Terms

• Check for terms with matching variables. It's important that both the variable and its exponent are the same.
• These matching terms are what you will gather together by combining their coefficients.

Recognizing like terms allows you to rewrite the expression in a simpler form by consolidating them into a single term, as shown with \(-8b - 9b = -17b\).