Problem 68 - Intermediate Algebra

Question

Translate each English phrase into an algebraic expression and use \(n\) to represent the unknown number. A number subtracted from 75

Step-by-Step Solution

Answer

The algebraic expression is \( 75 - n \).

Step 1: Identify the Operation

In the phrase 'a number subtracted from 75', 'subtracted from' indicates a subtraction operation. This means we will need to subtract the unknown number from 75.

Step 2: Identify the Unknown

The phrase mentions 'a number,' which is the unknown value. We are asked to represent this unknown with the variable \( n \).

Step 3: Write the Algebraic Expression

Taking into account the operation and the unknown, the phrase translates to the algebraic expression \( 75 - n \), where 75 is the initial value and \( n \) is the number being subtracted from it.

Key Concepts

subtraction in algebra

Subtraction in algebra is about understanding how to express the act of taking one quantity away from another mathematically. Unlike simple arithmetic subtraction where you're dealing with known numbers, algebra often involves subtracting unknowns or variables. In the example given, "a number subtracted from 75," we start with 75. The word "subtracted from" is crucial because it tells us that 75 is the starting point, and something is being taken away from it. This language signals us to perform a subtraction operation, which is written as:

• Start with the larger or initial number (here, 75).
• Subtract the unknown variable from it.

In algebra, the subtraction operation is straightforward once you identify what needs to be subtracted from what. Always pay attention to the wording of the phrase, as it dictates the structure of the expression.

unknown variables

An unknown variable in algebra is a letter or symbol used to represent a number that isn't known yet. In our example, the phrase "a number" is our unknown. We use variables to stand for unknown values in mathematical expressions or equations. It's a placeholder that can represent any number or, more generally, numbers satisfying certain conditions or equations.
In this exercise, "a number" is represented by the variable \( n \). This helps us build an expression or perform algebraic manipulations when the exact value is not provided. Using a variable:

• allows us to simplify problems by dealing with a single symbol rather than unknown numbers.
• enables us to express general ideas or formulas used for various values.

Knowing which variable to use can be based on convention (like using \( n \) in basic algebra problems), or determined by problem context.

translating phrases into expressions

Translating phrases into algebraic expressions involves converting words into mathematical language. This is a critical skill because it allows you to solve problems stated in everyday language. Let's go through the process using the phrase "a number subtracted from 75" as an example.Firstly, identify the quantities and operations mentioned:

• "75" is a specific number.
• "a number" is an unknown, represented by \( n \).
• "subtracted from" indicates subtraction.

Starting with the number 75, we interpret "subtracted from" as taking something away from 75, which leads to the expression \( 75 - n \). Note how the order of words aligns with the mathematical operation: the subtraction comes after we establish what we're taking away from (in this case, 75). This translation process helps turn real-world problems into solvable equations and expressions.