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Chapter 1: Problem 10

Simplify each expression. $$ -10-(7-14 r) $$

Short Answer

Expert verified
-17 + 14r

Step by step solution

01

- Distribute the Negative Sign

First, distribute the negative sign through the parentheses. This changes the sign of each term inside the parentheses. continue ewline -10 - (7 - 14r) = -10 - 7 + 14r
02

- Combine Like Terms

Now combine the like terms. The like terms here are the constant terms -10 and -7. ewline -10 - 7 + 14r = -17 + 14r

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Key Concepts

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

Understanding the Distributive Property
The distributive property is a fundamental concept in algebra that allows us to simplify expressions and equations. It states that multiplying a single term by a group of terms inside parentheses affects each term within those parentheses individually. In our example, we start with the expression:
\[ -10 - (7 - 14r) \]
The negative sign in front of the parentheses needs to be distributed to both 7 and \(-14r\). This results in:
\[ -10 - 7 + 14r \]
Notice that the negative sign in front of 7 and \(-14r\) has been applied, effectively changing their signs. Always remember, distributing often comes first in order of operations when simplifying algebraic expressions.
Combining Like Terms
Combining like terms is the process of simplifying expressions by adding or subtracting terms that have the same variables and exponents. In our example, after distribution, we have:
\[ -10 - 7 + 14r \]
Here, \(-10\) and \(-7\) are constant terms (they don't have variables), while \(14r\) is the variable term. To combine like terms, you can add or subtract them as follows:
\[ -10 - 7 = -17 \]
So our expression simplified becomes:
\[ -17 + 14r \]
This makes the expression simpler and easier to understand. Always look for similar terms in expressions and combine them to reduce the complexity.
Dealing with Negative Signs
Negative signs can sometimes be tricky to handle, but are essential in ensuring expressions are correctly simplified. In our example:
\[ -10 - (7 - 14r) \]
The negative sign outside of the parentheses means we need to change the signs inside the parentheses when we distribute it. This turns 7 into -7 and -14r into 14r:
\[ -10 - 7 + 14r \]
Negative signs also affect how you combine terms. For instance, combining \(-10 - 7\) involves considering both negatives to make a larger negative value, which is \(-17\). Being careful with negative signs will help prevent errors in simplifying expressions. Always distribute, then combine, keeping an eye on how negative signs change the terms.

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Most popular questions from this chapter

Perform each indicated operation. $$ \frac{-5(-6)}{9-(-1)} $$

Write a numerical expression for each phrase and simplify. The quotient of \(-20\) and the sum of \(-8\) and \(-2\)

Write each sentence as an equation, using \(x\) as the variable. Then find the solution from the set of integers between \(-12\) and \(12,\) inclusive. The quotient of a number and 4 is \(-1\)

Decide whether each statement is an example of the commutative, associative, identity, inverse, or distributive property. See Examples \(1,2,3,5,6,7,\) and \(9 .\) $$6(5 t)-6(7 r)=6(5 t-7 r)$$

Write a numerical expression for each phrase and simplify. The product of \(-\frac{1}{2}\) and \(\frac{3}{4},\) divided by \(-\frac{2}{3}\)
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