Problem 9 - Prealgebra

Question

Use the distributive property to combine each of the following pairs of similar terms. $$2(3 a+7)$$

Step-by-Step Solution

Answer

The expression simplifies to \(6a + 14\).

Step 1: Identify the components

In the expression \(2(3a + 7)\), we have a number \(2\) multiplied by a binomial \(3a + 7\). We will use the distributive property to multiply the \(2\) by each term inside the parentheses.

Step 2: Apply the distributive property

The distributive property states that \(a(b + c) = ab + ac\). Thus, apply it to \(2(3a + 7)\) by multiplying \(2\) by \(3a\) and \(2\) by \(7\).

Step 3: Perform the multiplication

Multiply \(2\) by \(3a\) to get \(2 \times 3a = 6a\). Also, multiply \(2\) by \(7\) to get \(2 \times 7 = 14\).

Step 4: Combine the results

Add the results of the multiplication: \(6a + 14\). This is the expression with the terms combined using the distributive property.

Key Concepts

Understanding Multiplication in Expressions

Multiplication is a fundamental mathematical operation, and in algebra, it involves combining numbers and variables. When you multiply terms like in the exercise, you are essentially scaling a value by another. In the expression \(2(3a + 7)\), you perform multiplication by distributing the number 2 to each term inside the parentheses.
This means:

• You multiply 2 by \(3a\), which yields \(6a\).
• You then multiply 2 by 7, resulting in 14.

Applying multiplication in this way allows you to simplify expressions and solve equations more effectively. It's a vital skill in algebra, helping to manage and manipulate expressions to find solutions. Remember, each term is treated separately in the multiplication process before summing them up.

Exploring Binomial Expressions

Binomial expressions are algebraic expressions containing two terms. The example \(3a + 7\) is a binomial expression because it consists of two distinct terms: \(3a\) and 7.
Let's break this down:

• The first term, \(3a\), includes a coefficient (3) and a variable part (a). This term can represent varying values depending on the input of the variable.
• The second term is a constant, 7, which remains unchanged since it does not contain any variables.

Binomial expressions are common in algebra. They often appear in situations where relationships between variables are being expressed. Understanding how to identify and manipulate binomial expressions will greatly enhance your ability to simplify and solve algebraic equations.

Combining Similar Terms

In algebra, similar terms are those that have the same variable raised to the same power. These terms can be combined to simplify expressions. In the exercise where we have reached the expression \(6a + 14\), there's no further combining of similar terms needed because:

• \(6a\) is a term with the variable 'a', while 14 is a constant.
• Since these terms do not share the same variable structure, they aren't similar and thus cannot be combined further.

When dealing with expressions, always look for terms with the same variable factor to combine. This process helps in simplifying the expression to its minimal form, which is crucial for solving equations efficiently. Combining similar terms reduces clutter and aids in better comprehension of the algebraic problem.