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Chapter 1: Problem 50

Use the distributive property to find the product. See Example II. $$ -(c+7 d) $$

Short Answer

Expert verified
The product is \(-c - 7d\).

Step by step solution

01

Identify the Expression to Distribute

The given expression is \(-1(c + 7d)\). Here, we need to distribute the \(-1\) through the terms inside the parentheses \((c + 7d)\).
02

Distribute Across the First Term

Apply the distributive property by multiplying \(-1\) by the first term, \(c\). This gives:\[-1 \cdot c = -c\].
03

Distribute Across the Second Term

Continue applying the distributive property by multiplying \(-1\) by the second term, \(7d\). This gives:\[-1 \cdot 7d = -7d\].
04

Combine the Results

Combine the results from Steps 2 and 3. The final expression after distribution is:\(-c - 7d\).

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Key Concepts

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

Understanding Algebraic Expressions
An algebraic expression is a combination of numbers, variables, and operators (like addition, subtraction, multiplication, and division) that represents a mathematical phrase. In our exercise, the expression is
  • \(c + 7d\)
This expression is composed of a variable \(c\) and another term \(7d\), where \(d\) is a variable and 7 is called a coefficient because it is a number that multiplies the variable \(d\).
The goal is often to simplify or manipulate these expressions, such as when using the distributive property to expand or combine terms. Understanding algebraic expressions is key to solving equations and performing operations effectively.
Working With Negative Numbers
Negative numbers are numbers that are less than zero and are often represented with a minus sign (-). In algebra, handling negative numbers is crucial because it affects how we perform operations, such as addition, subtraction, multiplication, and division.
When we have an expression like
  • \(-1(c + 7d)\)
we are using the concept of negative numbers to indicate that the entire expression within the parentheses needs to be multiplied by -1. This changes the sign of each term inside the parentheses.
So, when you see
  • \(-c\)
  • and \(-7d\)
you understand these are the results of multiplying a positive number by -1, demonstrating how signs are distributed and change in algebra.
Exploring Multiplication in Algebra
Multiplication in algebra works similarly to multiplication with regular numbers, but it involves combining constants (numbers) and variables. It is an essential skill for manipulating algebraic expressions and equations. Some key points to remember include:
  • Coefficients should be multiplied as usual numbers.
  • Variables are multiplied by joining them together (e.g., \(a \cdot b = ab\)).
In the example
  • \(-1(c + 7d)\)
We characterize this multiplication by distributing \(-1\) to each term:
  • First, \(-1 \cdot c = -c\)
  • Then, \(-1 \cdot 7d = -7d\)
These steps demonstrate how multiplication in algebra allows us to alter expressions while maintaining their equivalent value.

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Most popular questions from this chapter

Name the only real number that is its own opposite, and explain why this is so.

Write each of the following as an algebraic expression. See Examples 12 and 13. Write an expression for the amount of money (in dollars) in \(n\) nickels.

Write each of the following as an algebraic expression. See Examples 12 and 13. If \(x\) is an odd integer, represent the next odd integer as an expression in \(x\).

Evaluate each expression when \(x=9\) and \(y=-2 .\) See Example 12 $$ -7 y^{2} $$

Simplify each expression. See Examples \(11,14,\) and \(15 .\) $$ -(n+5)+(5 n-3) $$
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