Problem 56 - Elementary Algebra

Question

Use the commutative properties of addition and multiplication to write equivalent expressions for the following problems. $$ y+12 $$

Step-by-Step Solution

Answer

Answer: The equivalent expression is \(12 + y\).

Step 1: Identify the commutative property to apply

Since the given expression involves addition, we will use the commutative property of addition.

Step 2: Apply the commutative property of addition

According to the commutative property of addition, we can rewrite the given expression by swapping the terms: \(y + 12 = 12 + y\)

Step 3: Write the equivalent expression

Using the commutative property of addition, we found that the equivalent expression for the given problem is \(12 + y\).

Key Concepts

Algebraic Expressions

An algebraic expression is a combination of variables, numbers, and operations. These expressions are vital in algebra because they represent mathematical relationships concisely. For instance, in the expression \(y + 12\), \(y\) represents a variable that can take any value, while 12 is a constant number.

Algebraic expressions can be simple, like \(y + 12\), or more complex, involving multiple variables and operations. Understanding and manipulating these expressions is essential for solving problems in algebra. The task of rewriting \(y + 12\) as \(12 + y\) demonstrates an essential skill in algebra: recognizing how changes in structure leave values unchanged. It also highlights the simplicity and versatility of mathematical expression.

When working with algebraic expressions, remember that:

• Variables represent unknowns or quantities that can change.
• Constants are values that do not change.
• Operations describe how numbers and variables are combined.

Addition

Addition is one of the four fundamental arithmetic operations and is essential in both basic math and algebra. It involves combining numbers to find their total, or sum.

In algebra, addition is often used with variables, which may be represented in expressions like \(y + 12\). Here, the operation denotes combining the variable \(y\) with the constant 12.

A few key points about addition include:

• It is commutative, meaning that changing the order of the terms does not change the result.
• It is associative, which allows you to group terms in any order.
• It pairs easily with subtraction, its inverse operation.

Understanding addition as both an abstract concept and a useful tool helps simplify, compare, and solve algebraic expressions.

Mathematical Properties

Mathematical properties like the commutative and associative properties are rules that apply to numbers and operations, simplifying calculations and understanding. These properties set the groundwork for more advanced problem-solving in algebra.

• Commutative Property: States that the order of numbers in addition or multiplication doesn't affect the result. For example, \(y + 12 = 12 + y\).
• Associative Property: In addition or multiplication, the grouping of numbers doesn’t impact the outcome. For example, \((a + b) + c = a + (b + c)\).
• Identity Property: Describes that adding 0 or multiplying by 1 leaves any number unchanged. For example, \(y + 0 = y\).

These mathematical properties are useful shorthand that help reduce complexity in algebra. By applying these rules, expressions like \(y + 12\) are simplified into equivalent forms that offer the same results without changing values.