Problem 31 - Introductory and Intermediate Algebra for College Students 4th

Question

Write each English phrase as an algebraic expression. Let the variable \(x\) represent the number. nine subtracted from a number

Step-by-Step Solution

Answer

The algebraic expression for 'nine subtracted from a number' is \(x - 9\).

Step 1: Identifying the Elements

First, we identify the elements involved in the expression. In the phrase 'nine subtracted from a number', we have two core elements: 'nine', which is a numeral, and 'a number', which we are told is represented by the variable \(x\). 'Subtracted from' signifies a mathematical operation of subtraction.

Step 2: Interpreting the Phrase

Next, we interpret the English phrase into mathematical language. When subtraction is indicated, the number which follows 'from' is the number from which another number is being subtracted.

Step 3: Writing the Algebraic Expression

Finally, we can write our algebraic expression. The phrase 'nine subtracted from a number' is mathematically the same as 'a number minus nine'. So we represent it as \(x - 9\).

Key Concepts

Translating English to Algebra

Understanding how to convert English phrases into algebraic expressions is a fundamental skill in mathematics. It's like learning a new language where words are replaced by numbers and symbols. To master this, we must comprehend the meaning behind the words and the common ways they translate into algebraic terms.

For instance, when we encounter the term 'subtract' in a sentence like 'nine subtracted from a number,' we're dealing with a subtraction operation. The phrase highlights the order of operation as well: the number (which we'll call 'x') comes first, and 'nine' is subtracted from it. This indicates that the term following 'subtract' is the one being taken away. It's important not to rush through the phrasing. Take the time to visualize the situation: if you had some amount ('x'), how would you represent taking 'nine' away from it?

Key phrases to watch for include 'sum of,' indicating addition; 'difference,' pointing to subtraction; 'product,' for multiplication; and 'quotient,' for division. Translating such phrases correctly is crucial for setting up and solving algebraic equations effectively.

Variable Representation

In algebra, variables are symbols that stand in for unknown values. They are the placeholders that we can manipulate within expressions and equations to solve problems. Choosing an appropriate variable is the first step in transforming an English phrase into an algebraic expression.

For the phrase given in our exercise, 'a number' is quite vague in English, but in algebra, we represent this unknown quantity with a variable, often 'x'. The choice of the letter does not matter as long as it is consistent within the problem. When 'nine subtracted from a number' is encountered, we already have a predefined variable, 'x', to represent 'a number', making it our starting point for the expression.

Variables also help in forming relationships and patterns within math. If we had a series of numbers all being reduced by nine, we could represent this as 'x - 9', 'y - 9', 'z - 9', and so on, indicating that the same operation is happening to different numbers. The use of variables is essential for generalizing mathematical concepts and for tackling more complex problems.

Mathematical Operations

Mathematical operations are at the core of algebraic expressions and equations, and understanding them is critical. They include addition, subtraction, multiplication, and division, each with its own symbol and order of operation. When translating English to algebra, identifying these operations within a phrase transforms the spoken word into a precise mathematical directive.

For example, when we dissect the phrase 'nine subtracted from a number,' we are focused on the operation of subtraction, represented by the symbol '-'. The correct structure for this operation is to place the variable first and the number being subtracted second, resulting in 'x - 9'. It is also worth noting that the order in operations is significant, especially when the expression gets more complex involving multiple steps or operations. Always remember the BODMAS/BIDMAS rule (Brackets, Orders, Division/Multiplication, Addition, Subtraction) to determine the correct order in which to perform calculations.

In summary, recognizing and applying mathematical operations correctly is a skill that will support students not only in algebra but throughout their journey in learning mathematics.