Problem 47 - Beginning and Intermediate Algebra

Question

Simplify each expression. \(12 b+b\)

Step-by-Step Solution

Answer

13b

Step 1: Identify Like Terms

In the expression, identify the like terms. Here, both terms are like terms since they both have the variable 'b'. The terms are: 1) 12b 2) b

Step 2: Add the Coefficients

Combine the coefficients of the like terms. The coefficients are 12 and 1 (since 'b' is the same as 1b). This looks like: 12 + 1

Step 3: Write the Simplified Expression

After adding the coefficients (12 + 1), the simplified form of the expression is: 13b

Key Concepts

Like Terms

In algebra, we often need to simplify expressions. To do this, we first identify what we call 'like terms'.
Like terms are terms in an expression that have the same variable raised to the same power.
For example, in the expression \(12b + b\), both terms contain the variable 'b'.

Here are some key points about like terms:

• They must have the exact same variable(s).
• The variable(s) must be raised to the same power.
• Only like terms can be combined.

Identifying like terms helps us to simplify the expression more easily.

Coefficients

In algebraic expressions, the coefficient is the number that is multiplied by the variable.
For example, in the term \(12b\), 12 is the coefficient and 'b' is the variable.
Similarly, in the term \(b\), the coefficient is 1 (since \(b\) is the same as \(1b\)).

Understanding coefficients is crucial because:

• They tell us how many times the variable is being multiplied.
• They help us to easily combine like terms.
• They play a significant role in simplifying expressions.

Remember, coefficients are just numbers in front of variables that show multiplication.

Combining Terms

Once we've identified the like terms in an expression, the next step is combining them.
This involves adding or subtracting the coefficients of the like terms.
In the example \(12b + b\), we see that both terms are like terms because they both involve 'b'.

To combine them, simply add the coefficients. Here’s how you do it:

• Identify the coefficients (12 and 1).
• Add them together: \(12 + 1 = 13\).

The resulting term, after combining, is \(13b\).
This process helps us to write the expression in its simplest form.

Algebraic Simplification

Algebraic simplification is about making an expression easier to work with or understand.
Simplifying an expression involves reducing it to its most basic form.
This typically means combining all the like terms.

Here’s the process for simplifying \(12b + b\):

• First, identify like terms (which we did: both terms have 'b').
• Next, add the coefficients of these like terms (12 and 1).
• Finally, rewrite the expression with the combined coefficient (\(13b\)).

The simplified version of \(12b + b\) is \(13b\).
By simplifying expressions, we make them more concise and easier to solve or manipulate in further operations.