Problem 47 - Elementary Algebra

Question

For the following problems, rewrite each phrase using algebraic notation. 3 times \(x\) times \(y\)

Step-by-Step Solution

Answer

Question: Rewrite the phrase "3 times x times y" using algebraic notation. Answer: 3xy

Step 1: Identify the components

According to the given phrase, we need to deal with 3 main components which are: 1. "3" 2. "x" 3. "y" Our task is to multiply these three components together.

Step 2: Use algebraic notation for multiplication

We know that multiplication can be represented in algebraic notation simply by writing the components next to each other without any multiplication sign between them. So, the algebraic representation of an expression "3 times x times y" will be "3xy".

Key Concepts

Algebraic Expressions

Understanding algebraic expressions is fundamental in grasping algebra. These expressions are combinations of numbers, variables, and arithmetic operations (such as addition, subtraction, multiplication, and division). An essential aspect of algebraic expressions is knowing how to convert verbal phrases into algebraic terms.

As seen in the exercise where the phrase '3 times x times y' was given, the first step is to identify the quantities involved - which are the numbers and variables. Here, '3' is a constant, while 'x' and 'y' are both variables. The phrase essentially tells us to multiply these components together. By using algebraic notation, this multiplication does not require an explicit multiplication sign. Instead, the expression is written in a simplified form, such as '3xy'.

When transforming verbal phrases into algebraic expressions, it's important to maintain the order of operations and correctly represent constants and variables to ensure an accurate translation from words to mathematical language.

Multiplication in Algebra

Multiplication in algebra follows similar rules to multiplication in basic arithmetic; however, it simplifies the process by dropping the use of symbols such as the multiplication sign. When multiplying variables and constants in algebra, we simply place them side by side.

For example, multiplying '3' by 'x' by 'y' is elegantly written as '3xy’. However, students should note that the order in multiplication does not affect the product due to the commutative property of multiplication. This means that '3xy', '3yx', 'x3y', 'xy3', 'yx3', and 'y3x' all represent the same product.

To further understand multiplication in algebra, consider the associative property, which allows us to group terms in any way without changing the result. This property is especially useful when dealing with longer expressions, as it makes calculations more manageable and organized.

Variables in Algebra

In algebra, a variable is a symbol, usually a letter, that represents an unknown or a value that can change. Variables are one of the core elements of algebraic expressions and equations. They allow us to generalize statements and solve problems that apply to a wide range of possible values.

In the given exercise, 'x' and 'y' are used as variables. These stand-ins for values we do not yet know or values that can vary, allow us to perform operations just as we would with known numbers. It is crucial to understand that the values of variables can affect the overall solution, and different values of the variables will yield different results when substituted into the expression.

Variables can represent anything from simple numbers to complex expressions themselves, and they are integral to the abstraction and problem-solving power of algebra. They are the foundational building blocks that enable us to describe patterns, construct formulas, and solve real-world problems.