Problem 12 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ 7(3 a+2)+5 $$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(21a + 19\).

Step 1: Apply distributive property

First, apply the distributive property in the term \(7(3a + 2)\). This involves multiplying 7 with each term inside the bracket. So, \(7(3a + 2)\) = \(21a + 14\).The equation becomes \(21a + 14 + 5\).

Step 2: Combine the constant terms

Next, combine the constant terms in the equation. So, you add \(14 + 5\) to get 19. The final simplified expression is \(21a + 19\).

Key Concepts

Distributive Property

The distributive property is an essential tool in algebra that allows us to simplify expressions more easily. It involves multiplying each term inside a bracket by a factor outside the bracket. For example, in the expression \(7(3a + 2)\), the distributive property tells us to multiply 7 by each of the terms inside the parentheses: 3a and 2. Doing this gives us \(7 \times 3a\) and \(7 \times 2\).

Here's what happens step by step:

• First, multiply 7 by 3a to get 21a.
• Next, multiply 7 by 2 to get 14.

So, the distributive property simplifies \(7(3a + 2)\) to \(21a + 14\).

This method is very powerful and can be used to break down more complex expressions, making them easier to manage. Remember that this property is not only useful for numbers but also for algebraic expressions. Once you get used to it, the distributive property becomes your key to unlocking simplified forms of complicated expressions.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operation symbols. In algebra, we often work with these expressions to find solutions or simplify them. For instance, in the original exercise where we simplify \(7(3a + 2) + 5\), the expression includes:

• The number 7
• The variable term 3a
• The constant number 2
• Another constant number 5

The goal in simplifying algebraic expressions is to make them easier to understand and work with by reducing them to their simplest form. This might involve using the distributive property, combining like terms, or following order of operations. Each part of the expression plays a role in the final outcome, and understanding these parts is crucial for handling complex algebraic problems.

Algebraic expressions form the foundation in algebra, allowing us to express mathematical ideas clearly and concisely. Getting comfortable with them is a stepping stone to mastering more advanced mathematical concepts.

Combining Like Terms

Combining like terms is another fundamental concept when simplifying algebraic expressions. Like terms are terms in an expression that have the same variable raised to the same power. In our exercise, the expression after applying the distributive property became \(21a + 14 + 5\).

Here, 14 and 5 are the like terms because they are constants (they have no variables attached). The process to combine them is straightforward:

• Add 14 and 5 together to get 19.

Now, the expression simplifies further to \(21a + 19\).

By combining like terms, we reduce the number of terms in the expression, making it simpler and more concise. This is a crucial step in the simplification process, and it helps ensure that we end up with the most compact form of an expression. This technique, paired with others like distributing, forms a powerful set of tools students use in solving and simplifying algebraic expressions.