Problem 368 - Intermediate Algebra

Question

In the following exercises, simplify using the Distributive Property. $$ 9-6(7 n-5) $$

Step-by-Step Solution

Answer

39 - 42n

Step 1: Identify the terms

The expression given is: 9 - 6(7n - 5)

Step 2: Distribute -6

Apply the Distributive Property by multiplying -6 with each term inside the parentheses:-6 * 7n and -6 * -5.

Step 3: Perform the multiplication

Calculate the results of the multiplication:-6 * 7n = -42n-6 * -5 = 30.Thus, we have:9 - 42n + 30

Step 4: Combine like terms

Combine the constant terms (9 and 30):9 + 30 = 39.The simplified expression now is:39 - 42n

Key Concepts

Simplifying Expressions

In algebra, simplifying expressions involves reducing them to their simplest form. This process helps to make problems easier to solve.

The Distributive Property is a key tool in simplifying expressions. The property states that for any numbers or expressions a, b, and c, the following holds true: ewline a(b + c) = ab + ac.

Here's a step-by-step look at simplifying an expression using the distributive property:

• Start with the expression 9 - 6(7n - 5).
• Apply the Distributive Property: multiply -6 by both terms inside the parentheses.
• Perform the multiplication to get -42n and 30.
• The expression now looks like this: 9 - 42n + 30.
• Finally, combine like terms (the numbers 9 and 30) to simplify further.

Simplifying algebraic expressions becomes much more manageable when you use the distributive property effectively.

Combining Like Terms

Combining like terms is a fundamental aspect of simplifying algebraic expressions. Like terms are terms that have the same variable raised to the same power.
For example, in the expression 9 - 42n + 30, 9 and 30 are like terms because they are both constant terms. Here’s how to combine them:

• Identify the like terms: In this exercise, the like terms are 9 and 30.
• Add or subtract the constants as needed: 9 + 30 = 39.

After combining, you get: 39 - 42n.
Combining like terms simplifies the expression and makes it easier to work with. This is an essential skill for intermediate algebra.

Intermediate Algebra

Intermediate algebra builds on basic algebra concepts and introduces more complex topics. Understanding how to simplify expressions and combine like terms is crucial at this level.

In intermediate algebra, you will often encounter expressions that require you to:

• Use the Distributive Property to redistribute terms.
• Combine like terms to simplify expressions.
• Handle more complex equations and inequalities.

For example, in the exercise 9 - 6(7n - 5), we used these principles to simplify the expression to 39 - 42n. Mastering these foundational skills will enable you to tackle more challenging algebra problems with confidence.