Problem 24 - Prealgebra

Question

Apply the associative property to expression, and then simplify the result. \((3 y+7)+8\)

Step-by-Step Solution

Answer

The simplified expression is \(3y + 15\).

Step 1: Understanding the Expression

The expression given is \((3y + 7) + 8\). We need to apply the associative property to this expression to group terms differently, without affecting the results due to the nature of addition.

Step 2: Applying the Associative Property

The associative property for addition states that the way terms are grouped in addition does not change the sum. So, we rewrite the expression as \(3y + (7 + 8)\). Here, we've regrouped the numbers \(7\) and \(8\) to take advantage of the associative property.

Step 3: Simplifying the Grouped Terms

Now, we simplify the grouped terms \((7 + 8)\). Calculating this, we get \(7 + 8 = 15\). Therefore, the expression now becomes \(3y + 15\).

Key Concepts

Algebraic Expressions

Algebraic expressions form the foundation of algebra and mathematics as a whole. An algebraic expression is a combination of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division.
In the expression \((3y + 7) + 8\), we can identify the components:

• The numbers 7 and 8 are known as constants.
• The letter \(y\) stands for a variable, representing an unknown value.
• The term \(3y\) is a product of the coefficient 3 and the variable \(y\).

When working with such expressions, the goal is often to simplify them or to solve for the variable. Understanding the structure of algebraic expressions is crucial for manipulating and solving them effectively.

Simplification of Expressions

Simplifying an expression involves performing operations to condense it into its simplest form. This process often makes the expression easier to work with in further calculations or problem solving.
In our example, after applying the associative property, the expression \((3y + 7) + 8\) becomes \(3y + (7 + 8)\).
Simplification occurs when we calculate the sum within the parentheses:

• First, we perform the addition \(7 + 8\), which yields 15.
• The expression is then simplified to \(3y + 15\).

Simplifying expressions is a vital skill in algebra, helping to unveil the basic structure and value of an expression with respect to its variables.

Properties of Addition

The properties of addition are useful tools in algebra that help simplify complex expressions and solve equations effectively. Among these, the associative property is particularly noteworthy.

• Associative Property: This property states that the way in which numbers are grouped in addition does not affect the sum. Mathematically, this is expressed as \((a + b) + c = a + (b + c)\).

In our exercise, applying the associative property allowed us to regroup the constants, changing \((3y + 7) + 8\) into \(3y + (7 + 8)\), without changing the overall sum.
The properties of addition, like the commutative and associative properties, provide powerful techniques for reordering and simplifying expressions, proving to be fundamental in algebraic computations.