Problem 64 - Prealgebra

Question

Apply the distributive property, then simplify. $$\frac{3}{7}\left(\frac{7}{9} x-21\right)$$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{1}{3} x - 9 \).

Step 1: Understand the Expression

The problem is to simplify the expression \( \frac{3}{7}\left(\frac{7}{9} x - 21\right) \) by using the distributive property. This means that we need to distribute the \( \frac{3}{7} \) to each term inside the parentheses.

Step 2: Distribute \( \frac{3}{7} \) to \( \frac{7}{9} x \)

Apply the distributive property to the first term. Multiply \( \frac{3}{7} \) and \( \frac{7}{9} x \):\[\frac{3}{7} \times \frac{7}{9} x = \frac{3 \times 7}{7 \times 9} x = \frac{3}{9} x = \frac{1}{3} x.\]

Step 3: Distribute \( \frac{3}{7} \) to \(-21\)

Now, apply the distributive property to the second term. Multiply \( \frac{3}{7} \) and \(-21\):\[\frac{3}{7} \times (-21) = (-21) \times \frac{3}{7} = \frac{-21 \times 3}{7} = \frac{-63}{7} = -9.\]

Step 4: Combine Both Results and Simplify

Combine the results from Step 2 and Step 3:\[\frac{1}{3} x - 9.\] This is the simplified version of the expression.

Key Concepts

Simplifying Expressions

When simplifying expressions, the goal is to make the expression as simple as possible without changing its value. In algebra, this often entails reducing fractions, removing parentheses by distributing, and combining like terms. Let's break this down further.When you simplify an expression like \( \frac{3}{7}(\frac{7}{9} x - 21) \), you are working to make it less complex while maintaining its equivalency. This involves several important steps:

• Distribution: Every term inside the parentheses is affected separately. That means multiplying \( \frac{3}{7} \) by each of those terms one at a time.
• Combining Results: After distributing, you collect the results together to form a new, simpler expression.
• Reducing Fractions: Any resulting fractions should be reduced to their simplest form.

It's important that the simplified expression accurately reflects the original one, just in a more workable form, like we did to get \( \frac{1}{3} x - 9 \). This version is easier to use for solving further algebraic problems.

Fractions

Fractions represent parts of a whole and are a crucial aspect of simplifying algebraic expressions. When working with fractions in algebra:- **Multiplying Fractions**: You multiply the numerators together to get a new numerator, and the denominators together to get a new denominator, as shown by multiplying \( \frac{3}{7} \times \frac{7}{9} \).- **Simplifying Fractions**: Always check if the fraction can be reduced. Simplifying \( \frac{3}{9} \) to \( \frac{1}{3} \) is a good example.- **Converting Whole Numbers to Fractions**: When dealing with whole numbers, like \(-21\), convert them to fractions (such as \( \frac{-21}{1} \)) to ease the multiplication process with another fraction.In the given example, fractions enable us to systematically apply the distributive property, making complex calculations more manageable, thus allowing algebraic manipulation to focus on simplification.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables (like \( x \)), and operations such as addition, subtraction, multiplication, and division. Understanding how to work with these expressions is key to solving many mathematical problems.Here's what makes them special:

• **Variables**: They are symbols that represent unknown values or can change within a problem.
• **Operations**: You must be familiar with using addition, subtraction, multiplication, and division to manipulate expressions.
• **The Distributive Property**: This particular property is fundamental in breaking down and simplifying expressions, as it allows distribution of a factor across terms inside parentheses.

The core of working with algebraic expressions lies in maintaining balance and equivalency when manipulating them. In our step-by-step simplification example, we were able to distribute and simplify the expression \( \frac{3}{7}(\frac{7}{9} x - 21) \) to become \( \frac{1}{3} x - 9 \), demonstrating both the power and simplicity that comes from mastering algebraic operations.