/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 62 Simplify each algebraic expressi... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 62

Simplify each algebraic expression by combining similar terms. $$-(y-2)-(y+4)-(y+7)-2(y+3)$$

Short Answer

Expert verified
The simplified expression is \(-5y - 15\).

Step by step solution

01

Distribute the Negative Signs

Rewrite the expression while distributing the negative signs in front of the parentheses: \[ -(y-2) = -y + 2 \] \[ -(y+4) = -y - 4 \] \[ -(y+7) = -y - 7 \] Combine these results to get: \[ -y + 2 - y - 4 - y - 7 \] Additionally, distribute the \(-2\) in \(-2(y+3)\) to get:\[ -2y - 6 \] So the full expression is: \[ -y + 2 - y - 4 - y - 7 - 2y - 6 \]
02

Combine Like Terms

Gather all the like terms in the expression. Focus on combining the terms with \(y\) and the constant terms: - For \(y\)-terms: - Combine \(-y, -y, -y,\) and \(-2y\) to get \(-5y\).- For constant terms: - Combine \(2, -4, -7,\) and \(-6\) to get \(-15\).So, the simplified expression is:\[ -5y - 15 \]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Combining Like Terms
To simplify an algebraic expression, the first step is often to combine like terms. This means grouping together terms that have the same variables raised to the same power. In our expression, we have variables all involving the same variable,
which makes them all like terms.
  • Identify the terms with the same variable: In the given expression, \(-y, -y, -y,\text{ and } -2y\)
  • Combine them by adding their coefficients: Combine \(-1y + (-1y) + (-1y) + (-2y)\) to get \(-5y\)
By combining them, you simplify your expression, which reduces potential calculation errors later on. Always double-check that you've grouped all similar variables together.
Distributing Negative Signs
In expressions with parentheses, distributing signs, especially negatives, is crucial. Negative signs before a parenthesis reverse the signs of each term inside, which affects how we proceed.
  • Take note of the negative signs: In our case, negative signs need to be applied to \(y-2,\) \(y+4,\) \(y+7,\) and \(y+3\)
  • Distribute the negative across the parentheses: This transforms \(y-2\) to \(-y + 2,\) \(y+4\) to \(-y - 4,\) and so on
Always perform this operation before combining like terms, as it ensures all signs are correctly accounted for and the correct expression is simplified. Remember,
distribution ensures expression simplification flows seamlessly.
Simplifying Expressions
Once distribution and combination of like terms are complete, we reach the crucial stage of simplifying the overall expression. Simplification means condensing the expression into its simplest form, making it easier to interpret or apply further operations.
  • Check your expression: Ensure like terms are grouped, and subtraction/negative distribution is correctly applied
  • Reduce the expression: For example, grouping all \(y\)-terms and constants, as detailed earlier, provides \(-5y - 15\)
Simplifying is a core part of working with algebraic expressions, helping tidy them up for clearer understanding or further mathematical manipulation. Revisit any steps where the simplification might seem too fast-paced, ensuring you grasp each stage thoroughly before moving forward.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Simplify each algebraic expression by combining similar terms. $$9 x+5 y-4 x-8 y$$

State the property that justifies each statement. For example, \(3+(-4)=(-4)+3\) because of the commutative property for addition. $$(-9)(17)=17(-9)$$

Simplify each algebraic expression by combining similar terms. $$-12 a+14 b-3 a-9 b$$

Simplify each algebraic expression and then evaluate the resulting expression for the given values of the variables. \(5(x-2)+8(x+6)\) for \(x=-6\)

Simplify each algebraic expression and then evaluate the resulting expression for the given values of the variables. \(5 x-7 y-9 x-3 y\) for \(x=-1\) and \(y=-4\)
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.