Problem 26 - Prealgebra

Question

Simplify the following expressions by combining similar terms. In some cases the order of the terms must be rearranged first by using the commutative property. $$9 a+1+2 a+6$$

Step-by-Step Solution

Answer

The expression simplifies to \(11a + 7\).

Step 1: Identify Like Terms

In the expression \(9a + 1 + 2a + 6\), we first identify like terms. Like terms contain the same variables raised to the same power. Here, \(9a\) and \(2a\) are like terms, and \(1\) and \(6\) are like terms.

Step 2: Rearrange Terms

Using the commutative property, rearrange the expression to group like terms together: \(9a + 2a + 1 + 6\). This helps to easily combine like terms.

Step 3: Combine Like Terms

Add the coefficients of the like terms. For the terms with \(a\), it's \(9 + 2 = 11\), so that becomes \(11a\). For the constant terms \(1\) and \(6\), it's \(1 + 6 = 7\). Thus, the expression simplifies to \(11a + 7\).

Key Concepts

Like Terms

Like terms in algebra are terms that have the same variables raised to the same power. They are the building blocks for simplifying expressions, as these are the only terms that can be combined. Let's look into what makes terms "like."

Here are some tips:

• Terms like \(9a\) and \(2a\) are considered like terms because they both contain the variable \(a\).
• Constant numbers, like \(1\) and \(6\), are also like terms because they have no variable attached.

Understanding like terms is crucial because only these terms can be combined to simplify an expression. If terms are not like terms, they must remain separate in the expression. You'll often find like terms in polynomials as well, which can then be added or subtracted depending on their coefficients.

Commutative Property

The commutative property is one of the foundational properties of arithmetic and algebra. It states that the order in which you add or multiply numbers does not change the result.

Let's see how this helps you with simplifying expressions:

• When simplifying expressions like \(9a + 1 + 2a + 6\), you can safely rearrange to \(9a + 2a + 1 + 6\).
• Re-ordering helps you to visually group like terms together.

By doing this, you can make the next step of combining like terms a lot easier. This property is very handy in simplifying expressions and making calculations simpler.

Combining Like Terms

After identifying and rearranging like terms, the next step is combining them. Once your like terms are grouped together thanks to the commutative property, you can add or subtract them easily.

Here’s how to do it:

• For terms with variables like \(9a\) and \(2a\), you combine them by adding their coefficients: \(9 + 2 = 11\), resulting in \(11a\).
• For constant terms like \(1\) and \(6\), you simply add them together to get \(1 + 6 = 7\).

Once combined, the expression \(9a + 1 + 2a + 6\) simplifies to \(11a + 7\). This method of combining like terms is essential in transforming and simplifying algebraic expressions.