Problem 30 - Elementary and Intermediate Algebra

Question

Use the associative law of addition to write an equivalent expression. $$ x+(2+y) $$

Step-by-Step Solution

Answer

(x + 2) + y

Step 1: Recognize the Associative Law of Addition

The associative law of addition states that \(a + (b + c) = (a + b) + c\). This means the grouping of numbers being added does not affect the sum.

Step 2: Identify the Parts of the Expression

In the given expression \(x + (2 + y)\), identify \a = x, \ b = 2,\ and\ c = y\.

Step 3: Apply the Associative Law

Rearrange the expression according to the associative law: \(x + (2 + y) = (x + 2) + y\).

Step 4: Write the Equivalent Expression

The equivalent expression, using the associative law, is \((x + 2) + y\).

Key Concepts

Addition

Addition is one of the fundamental operations in mathematics. It involves combining two or more numbers to get a total or sum. The symbol for addition is the plus sign (+). When you add numbers, the order in which you add them doesn't matter because of the commutative property. Similarly, how you group the numbers doesn't alter the sum due to the associative property.

Algebraic Expressions

An algebraic expression is a mathematical phrase that includes numbers, variables, and operation symbols. For example, in the expression \( x + (2 + y) \), \( x \), and \( y \) are variables, while 2 is a constant. Algebraic expressions do not have an equality sign (=) and therefore do not form equations. Algebraic expressions can be simplified using various properties of mathematics, such as the associative and commutative properties.

Equivalent Expressions

Equivalent expressions are expressions that represent the same value, even though they may look different. For instance, \( x + (2 + y) \) and \((x + 2) + y\) are equivalent expressions. The associative law of addition helps us understand that even if the grouping of terms changes, the overall value does not. Simplifying expressions and finding their equivalent forms can be very useful in solving algebra problems and understanding underlying mathematical relationships.

Mathematical Properties

Mathematical properties are rules that describe how numbers and operations behave. Two important mathematical properties in the context of the associative law of addition are:

• Associative Property of Addition: This property states that how numbers are grouped in addition does not affect their sum. Mathematically, it is expressed as \(a + (b + c) = (a + b) + c\).
• Commutative Property of Addition: This property indicates that the order in which numbers are added does not change the sum. It is written as \(a + b = b + a\).

Understanding these properties helps in simplifying complex algebraic expressions and solving equations efficiently.