/*! 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 107 Simplify each expression, if pos... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 107

Simplify each expression, if possible. $$ 6-4(-3 c-7) $$

Short Answer

Expert verified
The simplified expression is \(12c + 34\).

Step by step solution

01

Distribute the Negative Sign

To simplify the expression, start by distributing the -4 across the terms inside the parentheses. Multiply -4 by each of the terms within the parentheses, \(-3c\) and \(-7\). This gives us:\[-4(-3c) + (-4)(-7)\].
02

Simplify Each Multiplied Term

Now, multiply -4 by -3c which results in \(12c\), and multiply -4 by -7 which results in \(28\). Therefore, the expression becomes:\[6 + 12c + 28\].
03

Combine Like Terms

Next, combine the like terms. The constants 6 and 28 can be added together to simplify the expression further. Add these two numbers:\[6 + 28 = 34\].Thus, the simplified expression is:\[12c + 34\].

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.

Distributive Property
The Distributive Property is a fundamental algebraic principle. It helps us multiply a single term by each term inside a parenthesis. Imagine it as a way to "unlock" terms from parentheses so everyone gets multiplied.

In mathematics, the Distributive Property is framed as:
\( a(b + c) = ab + ac \).

This means you take the term outside the parentheses, like \(-4\) in our example, and multiply it by each term inside.
  • In the expression \(-4(-3c-7)\), the \(-4\) multiplies both \(-3c\) and \(-7\).
  • This operation gives us: \( -4(-3c) + (-4)(-7) \).
By applying the Distributive Property correctly, we transform expressions into simpler forms, ready to be further simplified.
Combining Like Terms
Combining like terms is like grouping similar actors on a stage. In algebra, terms with the same variable parts can be added or subtracted.

Like terms have the same variable raised to the same power, while different terms have different variables or degrees. For example, \(12c\) and \(34\) are unlike terms because one has a variable and the other does not.

In our expression, \(6 + 12c + 28\):
  • \(6\) and \(28\) are constants, hence like terms. They can be combined to \(34\).
  • \(12c\) stands alone because it has a different form.
By grouping and combining like terms, you simplify the expression to its most compact form, which in this case is \(12c + 34\).
Negative Sign Distribution
Distributing a negative sign is like using a flashlight to shine on everything hidden in the parentheses. It ensures each term gets the appropriate sign.

When distributing a negative sign, multiply by \(-1\), creating a mirror effect on each term's sign. Let's explore this principle using \(-4(-3c - 7)\):
  • A negative times a negative, \(-4\) and \(-3c\), results in a positive: \(12c\).
  • A negative times a negative, \(-4\) and \(-7\), also results in a positive: \(28\).
Each term in the parentheses changes sign when a negative number distributes over them. This strategic distribution allows for a correct simplification of complex expressions.

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 expression, if possible. $$ c^{3}-3 c^{2}+9 c+3 c^{2}-9 c+27 $$

Fill in the blank with \(>\) or \(<: 0.3 \quad \frac{1}{3}\)

Simplify by combining like terms. $$ 9 z-7-z-19 z $$

Simplify each expression, if possible. a. \(2(7 x) 5\) b. \(2(7 x+5)\)

$$ \text { Add: }-3+(-4)+(-5)+4+3 $$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

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.