Problem 43 - Prealgebra

Question

Apply the distributive property to each expression and then simplify. $$3(2 a+4)+7(3 a-1)$$

Step-by-Step Solution

Answer

The simplified expression is \(27a + 5\).

Step 1: Apply the Distributive Property to the First Term

Consider the expression \(3(2a + 4) + 7(3a - 1)\). Let's first distribute the 3 in the expression \(3(2a + 4)\). This means we multiply 3 by each term inside the parenthesis.\[3(2a) + 3(4) = 6a + 12\]

Step 2: Apply the Distributive Property to the Second Term

Now, distribute the 7 in the expression \(7(3a - 1)\). Multiply 7 by each term in the parenthesis.\[7(3a) - 7(1) = 21a - 7\]

Step 3: Combine Like Terms

Now that we've distributed the numbers, combine like terms from the expression. We have two terms with 'a' (\(6a + 21a\)) and two constant terms (\(12 - 7\)). \[6a + 21a + 12 - 7 = 27a + 5\]

Step 4: Write the Simplified Expression

The final expression after simplifying is \(27a + 5\). All like terms have been combined, and the expression is simplified.

Key Concepts

Algebraic Expressions

Algebraic expressions are the building blocks of algebra. They consist of numbers, variables, and operations (such as addition, subtraction, multiplication, and division). In our example, the expression is \(3(2a + 4) + 7(3a - 1)\). Here, the numbers 3 and 7 are coefficients, while \(2a\) and \(3a\) include the variable 'a'.

Algebraic expressions can be simple, consisting of just a single term, or more complex like the one here, with multiple terms and operations.

• Variables represent unknown values and are often denoted by letters such as 'a', 'x', or 'y'.
• Coefficients are numbers that multiply the variables. In our case, the coefficients are 6 and 21 after applying the distributive property.
• Constants are fixed numbers without variables. In the expression, these are 12 and -7 before combining like terms.

Combining Like Terms

When you have an algebraic expression, it often includes like terms. These are terms that contain the same variables raised to the same power. For example, \(6a\) and \(21a\) are like terms in the expression because both terms contain the variable 'a'.

Combining like terms is essential to simplifying algebraic expressions.

• Start by grouping all like terms together in the expression. This involves aligning them based on their variables and constants.
• Next, add or subtract the coefficients of like terms to simplify them. For the terms \(6a + 21a\), you simply add the coefficients (6 + 21) to get \(27a\).
• Similarly, combine constant terms by performing the arithmetic operation. In this case, the constant terms 12 and -7 are combined through subtraction to yield 5.

Once like terms are combined, the algebraic expression is in its simplest form.

Pre-Algebra

Pre-algebra forms the foundation of algebra, preparing students for deeper mathematical concepts. It covers basic operations with numbers and introduces the idea of using variables.

In pre-algebra, students learn to explore expressions such as \(3(2a + 4) + 7(3a - 1)\) by using the distributive property.

• The distributive property allows you to multiply a single term across terms within parentheses. This helps in breaking down complex expressions into simpler, manageable parts.
• Practice with the distributive property builds skills in manipulating expressions and equations, which is crucial for more advanced algebra.
• Pre-algebra concepts also emphasize the importance of simplifying expressions by combining like terms, an essential step in solving equations and inequalities in higher algebra courses.

By mastering pre-algebra concepts, students gain the confidence needed to tackle more complicated algebraic problems.