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Chapter 1: Problem 141

Write an equivalent expression using an associative law. $$ (3 y+4)+10 $$

Short Answer

Expert verified
The equivalent expression is \3y + 14\.

Step by step solution

01

- Understand the associative law

The associative law states that the way in which numbers are grouped in an addition or multiplication operation does not change their sum or product. For addition, it can be written as \(a + (b + c) = (a + b) + c\).
02

- Identify the elements in the expression

In the given expression \((3y + 4) + 10\), the elements are \3y\, \4\, and \10\. The goal is to regroup these elements using the associative law.
03

- Apply the associative law

Using the associative law for addition, regroup the terms in the expression. Original: \((3y + 4) + 10\) can be written as \3y + (4 + 10)\.
04

- Simplify the grouped terms

Now, simplify the expression within the parentheses. \(4 + 10 = 14\). Therefore, \3y + (4 + 10) = 3y + 14\.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Associative Property
The associative property is a fundamental concept in algebra. It helps us understand that the grouping of numbers doesn't change the result of addition or multiplication. This means if you have an expression like \[ a + (b + c) \], you can regroup it as \[ (a + b) + c \] without altering the sum. This property is useful for simplifying calculations and solving algebraic expressions. In our example, we use the associative property to regroup \( (3y + 4) + 10 \) into \( 3y + (4 + 10) \). This reorganization makes it easier to simplify.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and operations. They allow us to represent real-world scenarios mathematically. For example, in the expression \( (3y + 4) + 10 \), \( 3y \) is a term containing the variable \ y \, and \ 4 \, \ 10 \ are constants. When working with algebraic expressions, it's important to understand the properties of operations such as addition, including the associative property.
Simplification
Simplification in algebra involves reducing expressions to their simplest form. This often means using properties like the associative property to regroup and combine terms. In our example, we started with \( (3y + 4) + 10 \). By applying the associative property, we regrouped it as \( 3y + (4 +10) \), then added the constants inside the parentheses to get \ 3y + 14 \.

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