Problem 10 - Elementary and Intermediate Algebra

Question

Complete each sentence using one of these terms: commutative, associative, or distributive. \(2(a+b)\) is equivalent to \(2 \cdot a+2 \cdot b\) by the _______ law.

Step-by-Step Solution

Answer

Distributive Law

Step 1: Understand the Problem

The problem asks to identify which law (commutative, associative, or distributive) explains the equivalence between the expressions \(2(a+b)\) and \(2 \cdot a + 2 \cdot b\).

Step 2: Review the Laws

Review the definitions of the laws: - Commutative Law: Order of addition or multiplication doesn't change the result, e.g., \(a + b = b + a\) or \(a \cdot b = b \cdot a\).- Associative Law: Grouping of addition or multiplication doesn't change the result, e.g., \( (a + b) + c = a + (b + c) \) or \( (a \cdot b) \cdot c = a \cdot (b \cdot c) \).- Distributive Law: Multiplication distributed over addition, e.g., \(a(b + c) = ab + ac\).

Step 3: Identify the Law in the Expression

Analyze the given expressions: \(2(a+b)\) as the original and \(2 \cdot a + 2 \cdot b\) as the equivalent form. Recognize that the multiplication \(2\) is distributed to both \(a\) and \(b\).

Step 4: Conclude the Correct Law

Conclude that the law used is the Distributive Law, as it distributes the multiplication across the addition inside the parentheses.

Key Concepts

commutative law

The commutative law is a fundamental property in algebra that states the order of addition or multiplication of numbers does not affect the result. For example, if you have two numbers, say 3 and 5, the commutative law tells us that:

• Addition: \(3 + 5 = 5 + 3\)
• Multiplication: \(3 \times 5 = 5 \times 3\)

This law is very handy when simplifying expressions or solving equations because it allows us to rearrange terms freely. When practicing with algebraic expressions, keep this property in mind as it often makes calculations easier and more intuitive.

associative law

The associative law deals with how numbers are grouped in addition or multiplication. It states that no matter how you group the numbers, the result will be the same. Here’s how it works:

• Addition: \((a + b) + c = a + (b + c)\)
• Multiplication: \((a \times b) \times c = a \times (b \times c)\)

This means that if you are adding or multiplying several numbers, you can change the grouping without changing the result. For instance, if you have \((2 + 3) + 4\), it is the same as \(2 + (3 + 4)\). Similarly, \((2 \times 3) \times 4\) is the same as \(2 \times (3 \times 4)\).

algebraic properties

Algebraic properties, like the commutative and associative laws, help us understand and manipulate mathematical expressions more easily. Here are some essential algebraic properties:

• Distributive Property: This property explains how multiplication distributes over addition or subtraction: \(a(b+c) = ab + ac\). This is the property used in the given exercise.
• Identity Property: This property states that adding 0 to a number doesn't change the number (\(a + 0 = a\)), and multiplying a number by 1 also doesn't change the number (\(a \cdot 1 = a\)).
• Inverse Property: For addition, every number has an additive inverse (opposite) that sums to zero (\(a + (-a) = 0\)). For multiplication, every non-zero number has a multiplicative inverse that multiplies to one (\(a \cdot \frac{1}{a} = 1\), for \(a eq 0\)).

Understanding these properties makes it easier to break down and solve complex algebraic problems by allowing you to rearrange, group, and simplify expressions systematically.