Problem 375 - Intermediate Algebra

Question

In the following exercises, simplify using the Distributive Property. $$ 5(2 n+9)+12(n-3) $$

Step-by-Step Solution

Answer

22n + 9

Step 1: Apply the Distributive Property to the first term

Distribute the 5 across the terms inside the first parenthesis:\[ 5(2n + 9) = 5 \times 2n + 5 \times 9 = 10n + 45 \].

Step 2: Apply the Distributive Property to the second term

Distribute the 12 across the terms inside the second parenthesis:\[ 12(n - 3) = 12 \times n + 12 \times (-3) = 12n - 36 \].

Step 3: Combine like terms

Add the results from the first and second distributions together:\[ 10n + 45 + 12n - 36 \].Group and combine the like terms (terms with \(n\) and constant terms):\[ (10n + 12n) + (45 - 36) = 22n + 9 \].

Key Concepts

Combining Like Terms

When you combine like terms, you are essentially simplifying algebraic expressions by grouping similar terms together. Like terms are terms that have the same variable raised to the same power. For example, in the expression \(10n + 12n\), both terms are like terms because they each have the variable \(n\) raised to the same power. To combine them, you simply add or subtract the coefficients (the numerical parts) of the terms. So, \(10n + 12n = 22n\). This process reduces the expression to a simpler form, making it easier to work with. Remember: coefficients are the numbers in front of the variables, and combining them helps you streamline your calculations.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves making the expression as concise as possible while retaining its original value. It often means performing operations like addition, subtraction, and distributing factors. For example, take the expression \(5(2n + 9) + 12(n - 3)\):

• First, use the Distributive Property to expand the expression: \(5 \times 2n + 5 \times 9 + 12 \times n + 12 \times -3 = 10n + 45 + 12n - 36\).
• Next, combine like terms to further simplify: \(10n + 12n + 45 - 36\).
• Finally, the simplified form is \(22n + 9\).

Always aim to write the simplest form to make further calculations easier.

Constants and Coefficients

In algebraic expressions, it's crucial to understand the roles of constants and coefficients. Constants are the numbers without variables. They stay the same, no matter what value the variables take. For instance, in the expression \(5(2n + 9) + 12(n - 3)\), 9 and -3 are constants.
Coefficients, on the other hand, are the numbers in front of the variables. They multiply the variables and determine the magnitude of the variable part of the term. In the same expression, the coefficients are 5 and 12 for the terms inside the parentheses.
Recognizing constants and coefficients is vital because it helps you apply the correct operations and simplify expressions efficiently. By understanding their roles, you can streamline solving and understanding algebraic problems.