Problem 86 - Beginning and Intermediate Algebra

Question

Use the distributive property to rewrite each expression. $$ -\frac{2}{5}(10 b+20 a) $$

Step-by-Step Solution

Answer

-4b - 8a

Step 1: Identify the expression to be distributed

The given expression is \[ -\left( \frac{2}{5} \right) (10b + 20a)\].

Step 2: Apply the distributive property

The distributive property states \[ a(b + c) = ab + ac.\] Here, distribute \[-\left( \frac{2}{5} \right)\] to both terms inside the parentheses. \[-\left( \frac{2}{5} \right) \cdot 10b + -\left( \frac{2}{5} \right) \cdot 20a\].

Step 3: Simplify each term

Calculate each term separately: \[-\left( \frac{2}{5} \right) \cdot 10b = -\left( \frac{2 \cdot 10}{5} \right)b = -4b\], and \[-\left( \frac{2}{5} \right) \cdot 20a = -\left( \frac{2 \cdot 20}{5} \right)a = -8a\].

Step 4: Combine the simplified terms

After simplifying both terms, combine the results to write the expression: \[-4b - 8a\].

Key Concepts

Distributive Property

The distributive property is a key arithmetic rule in algebra. It allows one to multiply a single term outside brackets to each term inside. This property is crucial for simplifying expressions and solving equations.
In algebra, the distributive property is written as \ a(b + c) = ab + ac. Here, you multiply the term \(a\) with every term inside the parentheses.
In our example, the term \(-\left( \frac{2}{5} \right)\) needs to be distributed to both \(10b\) and \(20a\). This ensures every term gets the multiplication it needs.
Understanding this property helps in breaking down more complex algebraic expressions into manageable parts. This ability to simplify is foundational for higher-level math.

Algebraic Expressions

An algebraic expression consists of numbers, variables, and operations (like addition or multiplication). Expressions can range from simple (e.g., \(2x+3\)) to complex (e.g., \(3ax - 5y + 7\)).
The main goal in working with expressions is to manipulate them so they are easier to understand and solve. This manipulation often involves using properties like the distributive property.
For instance, in the expression \(-\frac{2}{5}(10b+20a)\), you have variables (\(b\) and \(a\)), coefficients (\(-\frac{2}{5}\), \(10\), and \(20\)), and operations (multiplication and addition). Breaking down such expressions step by step aids in simplifying them correctly, making it easier to find solutions.

Simplifying Expressions

Simplifying expressions is about making them as straightforward as possible. This involves combining like terms, applying properties like the distributive property, and reducing fractions.
Here’s how we simplify the example expression:

• First, distribute the \(-\frac{2}{5}\): \ -\left( \frac{2}{5} \right) \times 10b + -\left( \frac{2}{5} \right) \times 20a\.
• Next, simplify each term: \ -\left( \frac{2 \times 10}{5} \right)b = -4b\ and \ -\left( \frac{2 \times 20}{5} \right)a = -8a\.
• Finally, combine the simplified terms: \ -4b - 8a\.

By simplifying, you turn complex expressions into more usable forms, making it easier for further calculations or solving equations.