Problem 22 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The expression without parentheses is \(-8y\).

Step 1: Understand the Expression

We have the expression \( \frac{4}{3}(-6y) \). The task is to rewrite this expression without parentheses by using the properties of real numbers.

Step 2: Apply the Distributive Property

To eliminate the parentheses, we apply the distributive property which states that \( a(b+c) = ab + ac \). In our expression \( \frac{4}{3} \) is multiplied by \( -6y \).

Step 3: Simplify the Multiplication

Multiply the fraction \( \frac{4}{3} \) with \( -6 \): \[ \frac{4}{3} \times -6 = \frac{4 imes -6}{3} = \frac{-24}{3} \]. Simplify \( \frac{-24}{3} \) to get \(-8\).

Step 4: Combine with Variable

After simplifying the multiplication of the numbers, combine it with the variable \( y \). This gives us the expression \(-8y\) without parentheses.

Key Concepts

Distributive Property

The distributive property is a key principle in algebra, very powerful when dealing with expressions that involve parentheses. It allows us to simplify expressions by distributing a factor over terms inside the parentheses. For example, if you have an expression like \( a(b + c) \), the distributive property tells us that it is equal to \( ab + ac \). This rule helps break down more complex expressions into simpler, easier-to-manage parts.

• Whenever you see an expression with a term outside of parentheses multiplied by terms inside, think of the distributive property.
• This property is particularly useful for expanding expressions or equations in algebra.

In our original exercise, we have \( \frac{4}{3}(-6y) \). By applying the distributive property, we treat \( -6y \) as a single entity and distribute \( \frac{4}{3} \) to it. This turns the original expression into a multiplicative operation of \( \frac{4}{3} \) and \( -6y \), letting us simplify it further.

Simplification

Simplification is the process of making an expression or equation as easy to handle as possible. It involves performing basic arithmetic operations and reducing expressions to their simplest form. In mathematics, simplifying helps to make problems and computations easier to interpret and solve.

• Always look for common factors that you can divide out from the numerator and the denominator.
• Be vigilant about negative signs and how they affect the gains from multiplication or division.

In our example, after applying the distributive property, we simplify the multiplication \( \frac{4}{3} \times -6 \). By performing the arithmetic, \( \frac{4 \times -6}{3} = \frac{-24}{3} \) simplifies to \(-8\). These steps ensure that we reduce the expression to the simplest form, making it easier to interpret and work with.

Variables

Variables in algebra are symbols used to represent unknown values or quantities that can change. Typically, letters from the alphabet such as \( x \), \( y \), or \( z \) are used as variables. Variables are crucial as they allow us to create generalizations and solve equations where some elements are unknown.

• Think of variables as placeholders in a mathematical sentence.
• They enable mathematicians and students to represent and work with numbers they do not yet know.

In our exercise, the letter \( y \) is the variable. Once we simplified the numeric part of our expression to \(-8\), the variable \( y \) remains. A simplified expression like \(-8y\) suggests "negative eight times the value of \( y \)," that offers a concise yet powerful representation of an expression involving a variable. Understanding how to manipulate variables is fundamental in algebra for solving equations and modeling real-world scenarios.