Problem 20 - Elementary and Intermediate Algebra

Question

Use the commutative law of multiplication to write an equivalent expression. $$ x y $$

Step-by-Step Solution

Answer

Using the commutative law, \( x \times y = y \times x \).

Step 1 - Understand the Commutative Law

The commutative law of multiplication states that the order in which you multiply two numbers does not affect the product. In other words, for any two numbers or variables, a and b, \[ a \times b = b \times a \].

Step 2 - Apply the Commutative Law to the Given Expression

Given the expression \( x \times y \), we can use the commutative law to rewrite it by swapping the two variables. This gives us \( y \times x \).

Key Concepts

Elementary Algebra

Elementary algebra is a branch of mathematics that deals with the basic concepts of algebra. It includes performing arithmetic operations using variables instead of numerals.
This forms the foundation for more advanced topics in mathematics. In elementary algebra, you will encounter the use of symbols (usually letters) to represent numbers.
These symbols, called variables, allow us to create generalized mathematical expressions and equations.

Multiplication

Multiplication is one of the basic operations in arithmetic and algebra.
It involves finding the product of two numbers or variables. When multiplying two variables, such as in the expression \( x \times y \), the commutative law of multiplication can be very useful.
The commutative law states that the order of multiplication does not matter:
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\( a \times b = b \times a \)

This means that you can rearrange the factors in a multiplication problem, and the result will stay the same. For example, \( x \times y \) can be rewritten as \( y \times x \), thanks to the commutative law.
This is particularly helpful when dealing with variables in algebra because it provides flexibility in how we approach and simplify expressions.

Variables

Variables are symbols, often letters, used to represent unknown or unspecified numbers in mathematical expressions and equations.
In the expression \( x \times y \), both \( x \) and \( y \) are variables.

• Variables can be manipulated according to algebraic rules.
• They allow us to generalize mathematical concepts and solve a wide range of problems.

For instance, through the commutative law of multiplication, we can change the order of variables in a multiplication expression without affecting the product.
Understanding how to work with variables is a crucial step in mastering elementary algebra and moving on to more complex algebraic concepts.