Problem 72 - Elementary Algebra Within Reach

Question

In Exercises 69 and 70, identify the variable(s) in the expression. $$ \frac{6}{t}-22 $$

Step-by-Step Solution

Answer

The variable in the expression \( \frac{6}{t}-22 \) is 't'.

Step 1: Identify the Expression

Look at the given expression: \( \frac{6}{t}-22 \).

Step 2: Identify the Variable

In the given expression \( \frac{6}{t}-22 \), the letter 't' is unknown, hence it is treated as a variable.

Step 3: Describe the Variable

The variable in the expression \( \frac{6}{t}-22 \) is 't'. It represents an unknown quantity that we could try to solve or use in further calculations.

Key Concepts

Identifying Variables

In algebraic expressions, **variables** are symbols or letters that represent unknown quantities. These are essential in mathematics because they allow us to create general formulas or solve problems using different values. In the expression \( \frac{6}{t}-22 \), the letter 't' is a variable.
When identifying a variable in an expression:

• Look for any letters or symbols which are not numbers.
• See how the variable is used in the expression (it might be part of a fraction, an exponent, a product, etc.).
  
• Understand that each variable can vary, meaning that it can take on different numerical values.

The context or the problem statement usually gives meaning to these symbols. Once a variable is identified, it becomes a tool for representing real-world situations or mathematical relationships.

Algebraic Expressions

An **algebraic expression** is a combination of numbers, operations, and variables. Expressions can include:

• Constants: Fixed numerical values (like 6 or 22 in the expression \( \frac{6}{t}-22 \)).
• Variables: Symbols that can change (such as 't' in this case).
  
• Operators: Mathematical symbols that denote operations (e.g., plus, minus, multiply, divide).
• Terms: Individual parts of the expression separated by plus or minus signs.
  

For the expression \( \frac{6}{t}-22 \):

• The term \( \frac{6}{t} \) involves a variable in a division operation.
• "-22" is a constant term being subtracted from \( \frac{6}{t} \).

Algebraic expressions play a crucial role in developing equations and functions, offering a way to model problems and make predictions.

Unknown Quantities

In mathematics, **unknown quantities** are values that need to be found or solved. They are often denoted by variables like 't' in the expression \( \frac{6}{t}-22 \). Understanding unknown quantities is vital as they form the basis of problems in algebra.
When dealing with unknown quantities, remember:

• They represent specific numbers that solve an equation or make a statement true.
• You often work to "solve for" these quantities, which means finding the values of the variables that make the expression or equation valid.
  
• In practical applications, these values often represent measurable items, such as time, distance, or quantities in a recipe.

Unknown quantities in expressions like \( \frac{6}{t}-22 \) allow us to pose and solve a wide variety of mathematical questions, helping us tackle challenging or real-world problems by finding what these variables could possibly stand for.