Problem 71 - Elementary Algebra

Question

Use the commutative property of multiplication to write a number equal to the number \(y x\).

Step-by-Step Solution

Answer

Question: Using the commutative property of multiplication, rewrite the expression yx in another form. Answer: xy

Step 1: Identify the numbers involved in the multiplication

In this case, the two numbers being multiplied are \(y\) and \(x\).

Step 2: Apply the commutative property of multiplication

Using the commutative property, we can rewrite the expression \(yx\) as \(xy\), since \(yx = xy\).

Key Concepts

The Basics of Multiplication

Multiplication is a fundamental mathematical operation that involves combining equal groups of objects. At its core, multiplication is about adding the same number multiple times. For instance, if you have 3 groups of 4 apples, instead of adding 4 + 4 + 4, you can simply multiply 3 by 4 to get 12 apples. This operation is represented in the form \(a \times b = c\), where \(a\) and \(b\) are the numbers being multiplied, and \(c\) is the product.

• The first number, \(a\), is called the multiplier.
• The second number, \(b\), is known as the multiplicand.
• The result, \(c\), is the product.

Understanding multiplication helps in solving various mathematical problems and is often used in everyday activities such as calculating total costs or measuring quantities.

Exploring Properties of Operations

Mathematical operations like multiplication have several properties that make calculations easier and understandable. One key property is the commutative property. This states that the order of the numbers does not change the product. For multiplication, this means \(a \times b = b \times a\). This property can significantly simplify calculations. For example, it's easier to compute \(5 \times 3\) instead of \(3 \times 5\) if you're more comfortable counting by fives.
Another important property is the associative property, which states that the way numbers are grouped does not affect their product. For instance, \((2 \times 3) \times 4 = 2 \times (3 \times 4)\). Lastly, there's the distributive property that combines addition and multiplication, allowing you to multiply a number by each addend separately and then add the results: \(a \times (b + c) = a \times b + a \times c\).
These properties are foundational in simplifying complex problems, helping students to approach mathematical calculations with greater ease.

Understanding Elementary Algebra

Elementary algebra is a branch of mathematics that introduces the use of symbols, often letters, to represent numbers. These symbols can stand for values that we may not know yet or can change, making algebra a powerful tool for solving equations and understanding relationships.
In algebra, we often use expressions like \(xy\) to indicate the multiplication of two variables \(x\) and \(y\). Thanks to the commutative property, this expression can also be written as \(yx\). This flexibility is helpful when solving equations or rearranging expressions to make them easier to understand or solve.

Examples in elementary algebra often include terms like \(3x + 5 = 20\). Here, \(x\) is a variable, and the goal is to find its value that makes the equation true. By understanding and using properties of operations, especially within algebra, you can simplify problems and accurately determine unknown values, which is a critical skill in both academics and real-life problem-solving.