Problem 69 - Elementary Algebra

Question

For the following problems, use the distributive property to expand the quantities. $$21.5(16.2 a+3.8 b+0.7 c)$$

Step-by-Step Solution

Answer

Question: Expand and simplify the expression using the distributive property: $$21.5(16.2 a + 3.8 b + 0.7 c)$$. Answer: $$348.3a + 81.7b + 15.05c$$.

Step 1: Identify the terms inside the parentheses

The given expression is $$21.5(16.2 a + 3.8 b + 0.7 c)$$. The terms inside the parentheses are 16.2a, 3.8b, and 0.7c.

Step 2: Apply the distributive property

Now, we'll multiply each term inside the parentheses by 21.5: $$21.5(16.2 a) + 21.5(3.8 b) + 21.5(0.7 c)$$.

Step 3: Calculate the products

Next, multiply 21.5 by each term: $$348.3a + 81.7b + 15.05c$$.

Step 4: Write the final simplified expression

The expanded expression using the distributive property is: $$348.3a + 81.7b + 15.05c$$.

Key Concepts

Algebraic Expressions

An algebraic expression is a combination of variables, numbers, and operations like addition, subtraction, multiplication, and division. They form the backbone of algebra and allow us to generalize mathematical ideas. When we have an expression like \(3(a+4)\), it contains a number 3 (known as a constant), and a variable "a". Here's how the components break down:

• Constants: Fixed values, like 3 in \(3(a+4)\).
• Variables: Symbols representing unknown values, such as “a” in \(3(a+4)\).
• Operations: These are the math procedures applied, such as multiplication and addition in \(3(a+4)\).

Expressions help solve equations and model real-world situations.

Expanding Expressions

Expanding expressions involves using the distributive property to multiply terms. The distributive property states that for any numbers \(a, b,\) and \(c\), \(a(b+c) = ab + ac\). This property allows you to remove parentheses by multiplying each term inside by the exterior term. Let's consider the expression \(21.5(16.2a + 3.8b + 0.7c)\):

• Identify the terms inside the parentheses: \(16.2a, 3.8b,\) and \(0.7c\).
• Multiply each term inside the parentheses by 21.5: \[ 21.5 \times 16.2a + 21.5 \times 3.8b + 21.5 \times 0.7c \]
• Calculate these products: \(348.3a, 81.7b,\) and \(15.05c\).
• Re-write the expression without parentheses: \(348.3a + 81.7b + 15.05c\).

Expanding expressions helps simplify complex structures into straightforward terms, making them easier to evaluate or solve.

Mathematical Problem Solving

Mathematical problem solving is a systematic approach to tackling problems using logical reasoning and appropriate mathematical techniques. It is essential for understanding complex concepts and applying them effectively. Here’s a structured approach:

• Understand the problem: Carefully read and determine what is being asked. For \(21.5(16.2a + 3.8b + 0.7c)\), identify terms inside the parentheses.
• Choose the best strategy: Decide whether to use a property or algorithm. Here, the distributive property is used.
• Execute the plan: Carry out the necessary calculations or transformations, like expanding the expression.
• Review your solution: Check the expanded expression to ensure calculations are correct.

By following these steps, one can efficiently solve mathematical problems, ensuring accuracy and understanding of the process.