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Chapter 1: Problem 377

In the following exercises, simplify using the Distributive Property. $$ 14(c-1)-8(c-6) $$

Short Answer

Expert verified
6c + 34

Step by step solution

01

- Apply the Distributive Property to the First Term

Distribute the 14 to both terms inside the parentheses: 14(c - 1) = 14c - 14
02

- Apply the Distributive Property to the Second Term

Distribute the -8 to both terms inside the parentheses: -8(c - 6) = -8c + 48
03

- Combine the Distributive Results

Combine the results from Step 1 and Step 2: 14c - 14 - 8c + 48
04

- Combine Like Terms

Combine the like terms (the terms with 'c' and the constant terms): (14c - 8c) + (-14 + 48) = 6c + 34

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Key Concepts

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

Simplification
The process of simplification in algebra involves reducing an expression to its simplest form. This often means applying mathematical operations like the Distributive Property and combining like terms. The aim is to make the equation easier to understand and solve.

Applying the Distributive Property: Let's break down the given problem. You are asked to simplify \(14(c-1)-8(c-6)\). First, we distribute the numbers outside the parentheses to each term inside the parentheses.
\[14(c - 1) = 14c - 14\]
\[-8(c - 6) = -8c + 48\]

By performing these operations, we break down the problem into smaller, simpler pieces. This makes each term clear and prepares us for the next step.
Combining Like Terms
Combining like terms means to merge terms that have the same variables and powers. This step further simplifies our expressions.

After distributing, we get:
\[14c - 14 - 8c + 48\]
We now look for terms that can be combined:
  • \(14c\) and \(-8c\)
  • \(-14\) and \(48\)
Combining like terms means doing the arithmetic on these grouped terms:
\[14c - 8c = 6c\]
\[-14 + 48 = 34\]

Putting these together, our simplified expression is \6c + 34.\
Intermediate Algebra
Intermediate algebra builds upon the basics you learn in elementary algebra. It involves more complex expressions and operations, like the ones we see here.

Understanding Intermediate Algebra: In this example, you are using intermediate algebra skills to simplify expressions through the Distributive Property and combining like terms. These skills are foundational for more advanced topics in algebra, such as solving equations and inequalities.

When working on problems of intermediate algebra:
  • Always break the problem into smaller steps.
  • Use properties like distribution to simplify each part.
  • Make sure to combine like terms to get the simplest form of the expression.
These techniques will make it easier to handle more complicated algebraic problems.

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Most popular questions from this chapter

In the following exercises, simplify using the Distributive Property. $$ -(3 x-7) $$

In the following exercises, locate the numbers on a number line. $$ \frac{3}{10}, \frac{7}{2}, \frac{11}{6}, 4 $$

Explain how you can multiply \(4(\$ 5.97)\) without paper or calculator by thinking of \(\$ 5.97\) as \(6-0.03\) and then using the Distributive Property.

In the following exercises, convert each decimal to a percent. $$ 3 $$

In the following exercises, locate the numbers on a number line. $$ \frac{7}{10}, \frac{5}{2}, \frac{13}{8}, 3 $$
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