Problem 59 - Algebra A Combined Function

Question

Write each algebraic expression described. In \(2009,\) the number of graduate students at the University of Texas at Austin was approximately 28,000 fewer than the number of undergraduate students. If the number of undergraduate students was \(n\), how many graduate students attend UT Austin? (Source: University of Texas at Austin)

Step-by-Step Solution

Answer

The number of graduate students is \( n - 28000 \).

Step 1: Understanding the problem

We need to express the number of graduate students at UT Austin in terms of a given variable, which is the number of undergraduate students, denoted by \( n \). The problem states that the graduate students are 28,000 fewer than the undergraduate students.

Step 2: Setting up the expression

Since the graduate students are 28,000 fewer than the undergraduate students, we subtract 28,000 from \( n \), the number of undergraduate students. The expression for the number of graduate students becomes \( n - 28000 \).

Step 3: Final expression

We have the expression for the number of graduate students at UT Austin as \( n - 28000 \). This equation represents the approximate number of graduate students.

Key Concepts

Variables in Algebra

In algebra, variables are symbols used to represent numbers, values, or amounts that are not yet known or need to be generalized. In our problem, the variable is denoted by \( n \). It acts as a placeholder for the number of undergraduate students at the University of Texas at Austin.

Variables are often represented by letters and can take on various values. They provide flexibility and a way to denote unknown quantities in a mathematical equation.

For example, if we know the number of undergraduate students, we can plug in this value for \( n \) to find the number of graduate students. This use of a variable helps in setting up a relationship between known and unknown quantities, reflecting the essence of what algebra aims to achieve. Understanding variables in algebra is crucial for forming and solving equations.

Subtraction in Algebra

Subtraction in algebra works much like subtraction in arithmetic. However, instead of only using numbers, we can subtract constants from variables. In the exercise, we subtract 28,000 from the variable \( n \) to find the number of graduate students.

This can be written mathematically as \( n - 28000 \). Here, \( n \) represents the total number of undergraduate students, and 28,000 is the given difference between undergraduate and graduate students.

Understanding subtraction in algebra involves knowing how constants and variables interact. With \( n - 28000 \), the result gives the distinct count of graduate students, emphasizing the amount fewer than the undergraduates they are. This ability to translate word problems into algebraic expressions and manipulate them is a foundational skill.

Understanding Algebraic Terms

Algebraic terms form the building blocks of expressions. In the given problem, the expression \( n - 28000 \) contains two terms: the variable term \( n \) and the constant term 28000. Identifying these components is crucial for solving equations.

Each term has its significance, with the variable term representing unknown quantities and the constant term signifying a fixed number. By identifying these, students can better understand how to construct and deconstruct algebraic expressions.

Recognizing algebraic terms allows us to interpret relationships and quantities effectively. As seen in our expression, the constant term 28000 shows reduction from the variable term \( n \), delivering a clear picture of the scenario being analyzed. By getting comfortable with identifying and manipulating these terms, solving algebraic problems becomes less intimidating and more intuitive.