Question

Identify the rule(s) of algebra illustrated by the statement.\(x(3 y)=(x \cdot 3) y=(3 x) y\)

Short answer

The given algebraic expression \(x(3 y)=(x \cdot 3) y=(3 x) y\) illustrates the Commutative and Associative Laws of Multiplication.

Step-by-step solution

Step 1. Identification of the Algebraic Laws

The first thing is to recognize the two important rules in play here. The given algebraic expression, \(x(3 y)=(x \cdot 3) y=(3 x) y\), can be separated into two parts comparing the first and second equations where \(x(3 y)=(x \cdot 3) y\), and the second and third equations where \((x \cdot 3) y=(3 x) y\). Start with examining the first part, where one can observe the order of the multiplication, did not change the outcome, that demonstrates the Commutative Law of Multiplication.

Step 2. Verification of Commutative Law

Commutative Law states that the order in which the numbers are multiplied does not matter. This rule is visible when comparing \(x(3 y)\) with \((x \cdot 3) y\). The 3 and y switched places, but the outcome remained the same. To ensure, you can test it using numbers. Suppose, if \(x = 2\) and \(y = 3\), then both sides yield the same result \(2 \cdot (3 \cdot 3) = (2 \cdot 3) \cdot 3\).

Step 3. Identification and Verification of Associative Law

Now, examine the second part of the given equation \((x \cdot 3) y = (3 x) y\). Here, the way the numbers are grouped changed, but the outcome has stayed the same, demonstrating the Associative Law of Multiplication. The Associative Law states that how you group the numbers when you multiply does not change the outcome. As the order of operations between x, 3, and y does not affect the result. Again, to back it with an instance, suppose \(x = 2\), \(y = 3\), then both sides yield the same result. \((2 \cdot 3) \cdot 3 = (3 \cdot 2) \cdot 3\).

Key concepts

Commutative Law of Multiplication

Multiplication may remind us of memorizing times tables, but there's more to it than just numeric gymnastics. The Commutative Law of Multiplication is a fundamental property that simplifies how we approach calculations. Think of it like this: When you have two numbers, let's say 'a' and 'b', and you multiply them together, it doesn't matter if you do 'a' times 'b' or 'b' times 'a', the answer—you guessed it—stays the same. Technically speaking, this means that for all numbers 'a' and 'b', the equation holds true: \( a \times b = b \times a \)
Imagine you're at a party and you exchange gifts with a friend—it doesn't matter who gives the gift first, the exchange happens either way. This law is especially helpful when working with algebraic expressions, where numbers and variables intermingle. It allows for the flexibility to rearrange terms to simplify an expression or solve for variables with greater ease.

Take the classroom scenario, for instance, where you have a group of 'x' students each bringing '3y' bags of supplies. According to the Commutative Law, multiply '3y' by 'x' or 'x' by '3y', and the total number of supply bags does not waver. Practical, isn't it?

Associative Law of Multiplication

While the Commutative Law plays with the order of factors, the Associative Law of Multiplication cares about the way numbers are grouped together. To get a glimpse of this concept, envision a trio of musical chairs labeled 'a', 'b', and 'c'. Regardless of which two chairs you first pair up for a duet, when you finally include the third, the musical harmony remains perfect. Similarly, \( (a \times b) \times c = a \times (b \times c) \).

Let's apply this to numbers. Suppose a baker has a recipe that calls for '2' batches of ingredients, with '3' cups of sugar and '4' eggs in each batch. She could group the sugar and eggs first, multiplying '3' by '4', and then by the '2' batches, or she could calculate '2' batches of sugar first and then multiply by the number of eggs. Either way, she needs the same total ingredients. The Associative Law affords us the convenience to group numbers in the most convenient format when tackling larger problems, which is a huge relief when squaring off against complex algebraic expressions.

Algebraic Expressions

When numbers and letters engage in a mathematical dance, we end up with algebraic expressions. These expressions are like recipes listing the ingredients (numbers, variables, and operators) without the cooking instructions (equals sign). For instance, an expression like \( 2x + 3y - 5 \) tells you what's in the mix but not what the final product is.

Understanding algebraic expressions is essential because it's the language of algebra. It's like learning the alphabet before writing stories. Dealing with algebraic expressions involves identifying terms, coefficients, variables, and constants. Think of terms as different dishes in a meal, coefficients as the quantity of each dish, variables as the dish types, and constants as the standard side dish that always comes with the meal—it never changes. Embracing algebraic expressions allows us to describe patterns, solve puzzles of unknown quantities, and understand the world of functions. And when combined with laws like associative and commutative, manipulating these expressions becomes a piece of cake—or should we say, a piece of pie 'π'?

Multiplication Properties

Dive into the realm of multiplication, and you'll discover an ocean of properties that guide how numbers interact with one another. Beyond the associative and commutative laws, there are others, such as the Distributive Property, which allows you to multiply a number by a sum of numbers within parentheses, spreading it across like butter on toast.

Realize that these properties do not exist to confuse but to clarify and streamline the problem-solving process. They are like the unwritten rules of board games that ensure fairness and predictability of play. Whether you are multiplying whole numbers, fractions, or algebraic terms, these properties ensure that the integrity of mathematics holds firm, like the steadfastness of a lighthouse against the mathematical waves. Embracing these multiplication properties can transform a daunting equation into a manageable and solvable formula.