Problem 85 - Beginning and Intermediate Algebra

Question

Use the distributive property to rewrite each expression. $$ -\frac{4}{3}(12 y+15 z) $$

Step-by-Step Solution

Answer

-16y - 20z

Step 1: Distribute the Coefficient

To apply the distributive property, multiply the coefficient \(-\frac{4}{3}\) by each term inside the parentheses. So, you will multiply \(-\frac{4}{3}\) by \(12y\) and \(-\frac{4}{3}\) by \(15z\).

Step 2: Multiply the Coefficient by the First Term

Multiply \-\frac{4}{3}\ by \12y:\ \[ -\frac{4}{3} \times 12y = -\frac{4 \times 12y}{3} = -\frac{48y}{3} = -16y \].

Step 3: Multiply the Coefficient by the Second Term

Multiply \(-\frac{4}{3}\) by \(15z\): \[-\frac{4}{3} \times 15z = -\frac{4 \times 15z}{3} = -\frac{60z}{3} = -20z \].

Step 4: Combine the Results

Combine the results of the multiplication: \(-16y - 20z\).

Key Concepts

Coefficient Multiplication

When working with algebraic expressions, multiplication of coefficients is a crucial step. Coefficients are the numerical parts of terms. Here, we need to distribute \(-\frac{4}{3}\) to both terms inside the parentheses: \(12y\) and \(15z\).

To perform coefficient multiplication:

• First, multiply the numbers: \(-\frac{4}{3} \times 12y\)
• Simplify by performing the multiplication: \(-\frac{48y}{3} = -16y\)
• Repeat with the second term: \(-\frac{4}{3} \times 15z\), simplifying to \(-20z\)

After completing the multiplications, you're left with values that replace the original terms in the parentheses.

Combining Like Terms

Combining like terms is another essential concept in algebra. Like terms are terms that contain the same variables raised to the same power. They can be combined by adding or subtracting their coefficients.

In the given expression, once you've applied the distributive property, you're left with \(-16y - 20z\). These terms are simplified as they do not have any like terms to combine.

Here's what you generally need to remember about combining like terms:

• Check for matching variables and exponents.
• Add or subtract only the coefficients of those like terms.
• Ensure any remaining terms are written next to each other without unnecessary changes.

Effective use of this concept simplifies complex expressions and makes problem-solving more manageable.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operations. They represent mathematical phrases that can describe real-world situations.

The original problem, \( -\frac{4}{3}(12y + 15z) \), is an example of using the distributive property in algebraic expressions.

Key points about algebraic expressions:

• Variables represent unknown values and can be part of any term in the expression.
• Operators such as addition, subtraction, multiplication, and division are used to describe relationships between numbers and variables.
• Practicing simplifying and manipulating expressions helps in solving complex equations later on.

Understanding these basics makes working with algebraic expressions much easier.