Problem 118 - Prealgebra

Question

Match each statement on the left with the property that justifies it on the right. a. Distributive property b. Associative property c. Commutative property d. Commutative and associative properties $$(a+4)+2=(4+2)+a$$

Step-by-Step Solution

Answer

The statement uses both the commutative and associative properties.

Step 1: Identify the Statement

The given statement is \((a+4)+2=(4+2)+a\). We need to identify which algebraic properties justify the equivalence of this expression.

Step 2: Analyze the Order Change

Observe that in the statement \((a+4)+2=(4+2)+a\), the order of terms changes. Originally, we have \(a+4\) grouped together, but after transformation, \(4+2\) is grouped and the position of \(a\) shifts. This indicates the use of the commutative property.

Step 3: Re-grouping the Terms

Notice that the grouping of terms is changed without altering their order inside the parentheses: \((a+4)+2\) changes to \((4+2)+a\). This is an application of the associative property which allows re-grouping.

Step 4: Conjunction of Properties

Since the expression involves both changing the order of terms and changing the grouping, both the commutative and associative properties are applicable here.

Key Concepts

Distributive Property

The distributive property is a fundamental property of algebra that helps us simplify expressions and solve equations. It describes how multiplication is distributed over addition (or subtraction). In mathematical terms, the property is expressed as:\[ a(b + c) = ab + ac \]This means if you are multiplying a number by a sum, you can "distribute" the multiplication to each individual term in the sum.For instance:

• In the expression \( 3(x + 4) \), you can distribute the \( 3 \) to both \( x \) and \( 4 \), resulting in \( 3x + 12 \).
• This property is useful when factoring or expanding expressions.

Understanding the distributive property helps in breaking down complex algebraic expressions and simplifying calculations.

Associative Property

The associative property is all about how you group numbers when you are adding or multiplying which does not change the result. This property deals solely with the grouping or parentheses used but not changing the order of the numbers themselves. In mathematical terms:- For addition: \( (a + b) + c = a + (b + c) \)- For multiplication: \( (ab)c = a(bc) \)Notice that the numbers themselves don't switch places, only the grouping changes.For example, if you have \( (2 + 3) + 4 \) or \( 2 + (3 + 4) \), both produce \( 9 \). This flexibility can ease computations without affecting results, by allowing you to group terms differently for convenience.

Commutative Property

The commutative property emphasizes that the order of numbers in addition and multiplication doesn't impact the result. This property allows you to rearrange terms with ease.In mathematical notation:- For addition: \( a + b = b + a \)- For multiplication: \( ab = ba \)In practical terms:

• With the expression \( 5 + 3 \), it holds that \( 5 + 3 = 3 + 5 \), which equals \( 8 \) in both cases.
• Similarly, for multiplication, \( 4 \cdot 2 = 2 \cdot 4 \), both yield \( 8 \).

The commutative property assures us that when adding or multiplying, rearranging terms doesn't affect the sum or product. This property is particularly useful in simplifying and solving equations.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and arithmetic operations. Unlike equations, they don’t have an equality sign. They're like phrases in the language of mathematics and can range from simple to complex.Components of algebraic expressions include:

• Variables like \( x \), \( y \), or \( a \), which can be placeholders for numbers or represent unknown values.
• Constants like numbers \( 2 \), \( 3.5 \), or negative values.
• Arithmetic operations: addition, subtraction, multiplication, and division.

Examples include: \( x + 2 \), \( 3y - 5 \), or \( 7x^2 - 4x + 6 \).Understanding algebraic expressions is crucial because they form the basis for formulating and solving equations, modeling real-world situations, and performing data analysis.