Problem 47 - Algebra 1

Question

DISTRIBUTIVE PROPERTY Use the distributive property to rewrite the expression without parentheses. $$ 5\left(\frac{1}{2} x-\frac{2}{3}\right) $$

Step-by-Step Solution

Answer

The expression without parentheses, by applying the distributive property, is \( \frac{5}{2}x - \frac{10}{3} \)

Step 1: Apply the Distributive Property

Use the distributive property to dissolve the parentheses by multiplying each term inside the parentheses by the factor outside. So, we get \( 5 \times \frac{1}{2}x - 5 \times \frac{2}{3} \)

Step 2: Simplify the terms

Now, perform the multiplication operations to simplify the obtained expression. This gives us \( \frac{5}{2}x - \frac{10}{3} \)

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. These expressions are like sentences in the language of algebra and they can represent real-world problems in a compact form. Unlike an equation, an algebraic expression doesn't have an equality sign. Take for example, the expression
\( 5\left(\frac{1}{2} x-\frac{2}{3}\right) \).
This contains numbers, a variable (\( x \)), parentheses, and multiplication, which are all fundamental components of an algebraic expression. Understanding how to work with these expressions is key to mastering algebra.

Simplifying Expressions

Simplifying expressions is an essential skill in algebra. It means reducing an expression to its most basic form without changing its value. This is where properties of operations come in handy. For the expression
\( 5\left(\frac{1}{2} x-\frac{2}{3}\right) \),
we first use the distributive property to expand the expression (that is, multiplying the 5 by each term inside the parentheses). Once expanded, we simplify any like terms or perform arithmetic to make the expression as concise as possible. The goal is to make the expression easier to understand and work with, which also reduces the risk of mistakes in further calculations.

Properties of Operations

The properties of operations are rules that apply to numbers and variables that allow us to manipulate and simplify expressions. They include the distributive property, associative property, commutative property, and the property of inverse operations among others. Specifically, the distributive property states that for any numbers a, b, and c,
\( a(b + c) = ab + ac \).
Applying this property, as seen in our example, makes it simpler to handle complex algebraic expressions. Understanding these properties is vital, as they are the backbone of algebraic manipulation, helping us solve equations and simplify expressions accurately and efficiently.