Problem 89 - Beginning and Intermediate Algebra

Question

Use the distributive property to rewrite each expression. $$ -3(8 x+3 y+4 z) $$

Step-by-Step Solution

Answer

-24x - 9y - 12z

Step 1: Identify the Distributive Property

The distributive property states that for any numbers a, b, and c: \( a(b + c) = ab + ac \). In this problem, the expression inside the parentheses is \((8x + 3y + 4z)\) and the number outside is -3.

Step 2: Apply the Distributive Property

Multiply -3 by each term inside the parentheses. This means: \(-3 \times 8x\), \(-3 \times 3y\), and \(-3 \times 4z\).

Step 3: Calculate Each Product

1. \(-3 \times 8x = -24x\) 2. \(-3 \times 3y = -9y\) 3. \(-3 \times 4z = -12z\).

Step 4: Combine the Results

Put all the products together: The rewritten expression is \(-24x - 9y - 12z\).

Key Concepts

Algebraic Expressions

An algebraic expression is a combination of numbers, variables (like x, y, z), and mathematical operations (like addition, subtraction, multiplication, and division). It's important to recognize that algebraic expressions do not have an equals sign—that would make them equations.
For example, the expression given in our exercise, \(8x + 3y + 4z\), is an algebraic expression. Each part, like \(8x\), \(3y\), and \(4z\), is called a term. Understanding how to manipulate these terms is crucial for solving more complex math problems. Being comfortable with terms and how they're combined will help you understand and solve algebra problems more effectively.

Multiplication

Multiplication in the context of algebra involves multiplying numbers and variables. When you multiply a number by a variable, you simply place them next to each other, such as \(-3 \times 8x = -24x \). Here, -3 is multiplied by 8x:

• The number -3 is a coefficient, and 8 is the coefficient for x.
• By multiplying, we combine these elements to get a new term: \(-24x\).
• This process is done similarly for other terms inside the parentheses, like \3y\ and \4z \.
In our exercise, we need to apply this to every term within the expression \ (8x + 3y + 4z) \:

• \-3 \times 8x = -24x\
• \-3 \times 3y = -9y\
• \-3 \times 4z = -12z\
Multiplying correctly will bring us closer to the final simplified expression.

Simplifying Expressions

Simplifying an algebraic expression means combining like terms and making it as compact and straightforward as possible. Once we've multiplied \-3\ with each term inside the parentheses, our expression changes.
In our exercise, \(-3(8x+3y+4z) \), we've calculated:

• \-3 \times 8x = -24x \
• \-3 \times 3y = -9y \
• \-3 \times 4z = -12z \

Now, we need to put these new terms together. The resulting expression is \-24x - 9y - 12z\. In this new form, there are no parentheses, and all terms are combined. This simplified expression is now in its most straightforward form and easily understandable and usable for future calculations. Remember, simplification makes solving algebra problems simpler and more direct.