Problem 18 - Algebra and Trigonometry

Question

\(15-20\) : Use properties of real numbers to write the expression without parentheses. $$ \frac{4}{3}(-6 y) $$

Step-by-Step Solution

Answer

The expression simplifies to \(-8y\).

Step 1: Identify the Expression

We have the expression \( \frac{4}{3}(-6y) \). Our task is to simplify this and remove any parentheses.

Step 2: Apply the Distributive Property

We identify that the expression \( \frac{4}{3}(-6y) \) is the multiplication of two terms: \( \frac{4}{3} \) and \(-6y \). According to the distributive property, we multiply \( \frac{4}{3} \) by \(-6y \): \[ \frac{4}{3} \times (-6y) \].

Step 3: Simplify the Multiplication

We perform the multiplication of the fractions by multiplying the numerators together and the denominators together:\[ \frac{4 imes (-6)}{3} imes y = \frac{-24}{3}y \].

Step 4: Reduce the Fraction

Next, simplify the fraction \( \frac{-24}{3} \) by performing the division, which gives us:\[ -8y \].

Key Concepts

Simplifying Expressions

Simplifying expressions is like organizing a messy desk. You want to make an algebraic expression as simple and straightforward as possible, without changing its value. When you simplify, you break down complex expressions into their simplest form.
This process often involves:

• Combining like terms, which are terms with the same variables raised to the same power.
• Reducing fractions to their simplest form.
• Using properties of operations, like the distributive property, to eliminate parentheses and condense expressions.

Simplification not only makes expressions easier to understand but also easier to work with in future calculations.

Distributive Property

The distributive property is one of the key tools for simplifying expressions. In math, it allows us to get rid of parentheses by distributing one operation across terms inside the parentheses.
For example, with an expression like \( a(b + c) \), you can distribute 'a' by multiplying it with both 'b' and 'c':

• \( a \times b + a \times c \)

This property becomes very useful when you have expressions like \( \frac{4}{3} \times (-6y) \), which can be rewritten by directly multiplying the fraction with everything inside the parentheses. The distributive property simplifies the expression, making it one step closer to its simplest form.

Fraction Multiplication

Multiplying fractions is often easier than it looks. You just need to multiply the numerators (the top parts) and the denominators (the bottom parts) of the fractions. Let's break it down a bit more:

• When you multiply fractions: \( \frac{a}{b} \times \frac{c}{d} \), the resulting multiplication would be \( \frac{a \times c}{b \times d} \).
• In our exercise, we multiply \( \frac{4}{3} \) by \(-6y\). Here, \(-6y\) can be viewed as \( \frac{-6y}{1} \). Thus, the multiplication becomes \( \frac{4 \times (-6y)}{3 \times 1} = \frac{-24y}{3} \).

By simplifying the resulting fraction, we make our expression easier to handle, especially when we encounter similar problems in the future.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operations. Unlike an equation, it doesn't have an equality sign. Simplifying algebraic expressions involves making them more concise while maintaining their meaning.
Examples of elements in algebraic expressions include:

• Variables: symbols that represent unknown values (like 'y' in the exercise).
• Constants: known fixed values (like the number 4 or -6 in our expression).
• Operations: addition, subtraction, multiplication, and division.

By learning how to manipulate these elements using rules and properties, such as combining like terms and applying the distributive property, one can effectively simplify complex expressions. This greatly aids in problem-solving scenarios where understanding and quick manipulation of algebraic expressions are required.