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Chapter 4: Problem 24

Translate into a variable expression. 19 less than the product of \(n\) and \(-2\)

Short Answer

Expert verified
The variable expression is \(-2n - 19\).

Step by step solution

01

Identify the Key Words and Phrases

The key words are '19 less than', 'product' and 'of', and n and -2. '19 less than' indicates a subtraction operation, 'product' indicates multiplication, 'of' also implies multiplication, and \(n\) and \(-2\) are the numbers being operated on.
02

Translate the Words to Mathematical Operations

The statement translates to a mathematical operation as follows: The 'product of \(n\) and \(-2\)' translates to \(-2n\), as 'product' indicates multiplication. Then, '19 less than' translates to subtracting 19 from this product, so it becomes \(-2n - 19\).
03

Write the Final Mathematical Expression

The final variable expression from the English statement '19 less than the product of \(n\) and \(-2\)' is \(-2n - 19\).

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Key Concepts

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

Variable Translation
When we talk about variable translation, we are essentially translating words or phrases into mathematical symbols or expressions. This is a key part of algebra. So, when given a phrase like "19 less than the product of \(n\) and \(-2\)", we start by identifying the variables and constants from the description.

Let's break it down:
  • The term "product of \(n\) and \(-2\)" suggests that a multiplication is involved. Here, \(n\) is our variable, and \(-2\) is a constant.
  • The phrase "19 less than" is crucial as it specifies that we need to perform a subtraction.
When translating such expressions, always start by identifying keywords and associate them with their mathematical meanings. This helps in crafting the correct mathematical equation.
Mathematical Operations
Mathematical operations are actions that take place between numbers or variables, such as addition, subtraction, multiplication, and division. They form the backbone of algebraic expressions.

In the given exercise:
  • The word "product" indicates the need for multiplication. Hence, the product of \(n\) and \(-2\) is represented as \(-2n\).
  • Then, the phrase "19 less than" identifies subtraction from this product.
Therefore, we directly translate the statement "product of \(n\) and \(-2\), 19 less than..." into \(-2n - 19\). Always ensure that operations are performed in the correct order to maintain mathematical accuracy.
Subtraction in Expressions
Subtraction can sometimes be tricky in algebraic expressions if not handled correctly. It involves taking one quantity away from another. In many expressions, the order in which you perform subtraction is crucial.

Here is how subtraction is applied in this scenario:
  • When you encounter "19 less than the product of \(n\) and \(-2\)", realize that it refers to taking 19 away from something (in this case, \(-2n\)).
  • This means subtracting 19 from the multiplication result, which gives us \(-2n - 19\).
Always pay attention to phrasing such as "less than" in algebraic contexts, as they often reverse the order of subtraction, which is a common source of errors.

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