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Chapter 1: Problem 19

In each term, give the numerical coefficient of the variable(s). See Objective 2. $$ -x $$

Short Answer

Expert verified
The numerical coefficient is \(-1\).

Step by step solution

01

Identify the variable

Observe the expression \(-x\). The variable in this term is \(-x\).
02

Determine the coefficient

In the expression \(-x\), the numerical coefficient is the number that multiplies the variable. If no number is explicitly shown, it is implied to be \(-1\).
03

Conclude the value

Thus, the numerical coefficient of the variable \(-x\) is \(-1\).

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Key Concepts

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

Variables in Algebra
In every algebraic expression, you'll find variables. A variable is a symbol, usually a letter, that stands for an unknown value. Imagine you're at a garage sale, and the price of an item isn't listed. The price is a variable because it's unknown until you ask the seller.

Variables help make generalized statements in math. For example, in the expression \( -x \), \ 'x' \ is the variable. It represents any number we might choose.

To solve algebraic problems, first, identify the variables. This will help you understand the problem and find solutions.
Algebraic Expressions
Algebraic expressions combine numbers and variables through operations like addition, subtraction, multiplication, and division. They are used to represent mathematical relationships in a general form.

For example, in the expression \( -x \), we combine a number (here it's -1) and a variable (\ 'x').

Let's dive deeper:
  • A term is a single part of an expression separated by addition or subtraction. In \( -x \), \( -x \) is a single term.
  • The numerical part of the term is called the coefficient. If the coefficient isn't shown, as in \( -x \), it is implied.
Breaking down algebraic expressions helps simplify problems and find solutions more easily.
Negative Coefficients
A coefficient is a number that multiplies a variable. Negative coefficients occur when the coefficient is negative.

In the expression \( -x \), the coefficient is -1 because it's the number multiplying the variable \'x'.

To identify the coefficient:
  • Look at the number in front of the variable. If there's no number, it's 1.
  • If there's a minus sign, the coefficient is -1.
Negative coefficients are like owing money. If you owe \(1, it's -\)1. In algebra, \( -x \) means you have -1 'x'.

Understanding negative coefficients helps in solving algebraic problems where signs can change the outcome.

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Most popular questions from this chapter

The two tables show the heights of some selected mountains and the depths of some selected trenches. Use the information given to answer. \( \)\begin{array}{|l|c|}\hline \text { Mountain } & {\text { Height (in feet) }} \\\ {\text { Foraker }} & {17,400} \\ {\text { Wilson }} & {14,246} \\\ {\text { Pikes Peak }} & {14,110} \\ \hline\end{array} $$ \begin{array}{|l|c|}\hline\text {} & {\text { Depth }( \text { in feet } } \\ {\text { Trench }} & {\text { as a negative number }} \\ {\text { Philippine }} & {-32,995} \\ {\text { Cayman }} & {-24,721} \\ {\text { Java }} & {-23,376} \\ \hline \end{array} $$ How much deeper is the Cayman Trench than the Java Trench?

A manufacturer has fixed costs of \(\$ 1000\) to produce widgets. Each widget costs \(\$ 5\) to make. The fixed cost to produce gadgets is \(\$ 750,\) and each gadget costs \(\$ 3\) to make. Write an expression for the cost to make \(y\) gadgets.

Solve each problem. Among entertainment expenditures, the average annual spending per U.S. household on fees and admissions was \(\$ 526\) in 2001 . This amount decreased \(\$ 32\) by 2003 and then increased \(\$ 112\) by \(2006 .\) What was the average household expenditure for fees and admissions in \(2006 ?\)

The two tables show the heights of some selected mountains and the depths of some selected trenches. Use the information given to answer. \( \)\begin{array}{|l|c|}\hline \text { Mountain } & {\text { Height (in feet) }} \\\ {\text { Foraker }} & {17,400} \\ {\text { Wilson }} & {14,246} \\\ {\text { Pikes Peak }} & {14,110} \\ \hline\end{array} $$ \begin{array}{|l|c|}\hline\text {} & {\text { Depth }( \text { in feet } } \\ {\text { Trench }} & {\text { as a negative number }} \\ {\text { Philippine }} & {-32,995} \\ {\text { Cayman }} & {-24,721} \\ {\text { Java }} & {-23,376} \\ \hline \end{array} $$ What is the difference between the height of Mt. Foraker and the depth of the Philippine Trench?

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. See Example \(9 .\) 7 less than a number is 2
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