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Chapter 1: Problem 142

Write an equivalent expression using an associative law. $$ x+(2 y+5) $$

Short Answer

Expert verified
(x + 2y) + 5

Step by step solution

01

Identify the given expression

The given expression is \(x + (2y + 5)\). Here, we have an addition operation involving a term and a parenthesis with an expression inside it.
02

Recall the Associative Law of Addition

The associative law of addition states that when adding three or more numbers or expressions, the way in which the numbers are grouped does not change their sum: \[a + (b + c) = (a + b) + c\]
03

Apply the Associative Law

In this step, apply the associative law to the given expression. According to the law: \[x + (2y + 5) = (x + 2y) + 5\]
04

Verify the Equivalent Expression

By grouping \(x\) and \(2y\) together first and adding 5 last, we ensure that the expression \((x + 2y) + 5\) is equivalent to the original expression \(x + (2y + 5)\).

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Key Concepts

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

algebraic expressions
An algebraic expression is a mathematical phrase that can contain numbers, variables (like x or y), and operation symbols like +, -, *, and /. An important thing to recognize about algebraic expressions is that they can be simplified or manipulated using various mathematical laws and properties. For example, in our given problem, the expression is x + (2y + 5). Here, x and y are variables, and (2y + 5) is a compound expression contained within parentheses. This expression stands as a perfect starting point for demonstrating the associative law of addition. Breaking down and understanding these components is key to mastering algebra.
parentheses in math
Parentheses play a crucial role in mathematics by indicating which operations should be performed first in an expression. They effectively group parts of an expression to ensure clarity and correct order of operations.
In our expression x + (2y + 5), the parentheses tell us that we should first consider the addition of 2y and 5 as one group before adding x. When applying the associative law, we effectively change the grouping of these terms without affecting the final result of the expression. Using parentheses correctly ensures that mathematical expressions are interpreted in the way they were intended.
addition properties
Addition properties are rules that make working with numbers easier and more intuitive. One of the most important properties is the associative law of addition. This law states that when you add three or more numbers, the way the numbers are grouped does not affect the sum.
In mathematical terms, this law is written as: \(a + (b + c) = (a + b) + c\).
Applying this to our problem, x + (2y + 5), we use the associative law to regroup the terms:
\(x + (2y + 5) = (x + 2y) + 5\).
This simply shows that the sum remains the same whether you add 2y and 5 first or add x and 2y first. Understanding this property helps in simplifying and solving algebraic expressions more effectively.

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Most popular questions from this chapter

Which numbers in the list provided are (a) whole numbers? (b) integers? (c) rational numbers? (d) irrational numbers? (e) real numbers?. $$ -8.7,-3,0, \frac{2}{3}, \sqrt{7}, 6 $$

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

To the student and the instructor: Synthesis exercises are designed to challenge students to extend the concepts or skills studied in each section. Many synthesis exercises require the assimilation of skills and concepts from several sections. On a quiz, Mia answers \(6 \in \mathbb{Z}\) while Giovanni writes \(\\{6\\} \in \mathbb{Z} .\) Giovanni's answer does not receive full credit while Mia's does. Why?

To the student and the instructor: Throughout this text selected exercises are marked with the icon Ahal. Students who pause to inspect an Ahal exercise should find the answer more readily than those who proceed mechanically. This may involve looking at an earlier exercise or example, or performing calculations in a more efficient manner. Some Ahal exercises are left unmarked to encourage students to always pause before working a problem. Evaluate each expression using the values provided. $$ m^{2}-[2(m-n)]^{2}, \text { for } m=7 \text { and } n=5 $$

Calculate using the rules for order of operations. If an expression is undefined, state this. $$ |-2-3| \cdot 4^{2}-3 $$
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