Problem 66 - Algebra 1

Question

You and your family take a summer vacation to Ireland. You discover that the number of Americans visiting Ireland is increasing by \(80,000\) visitors per year. Let \(x\) represent the number of visitors in 1997 . Write an expression for the number of visitors in 2000 .

Step-by-Step Solution

Answer

The expression representing the number of visitors in 2000 is \(y = 240000 + x\).

Step 1: Understanding the problem

We know that the number of Americans visiting Ireland increases by 80,000 each year. If \(x\) represents the number of visitors in 1997, we want to find an expression that represents the number of visitors in the year 2000.

Step 2: Determining the difference in years

We want to find the number of visitors in 2000 while we know the number of visitors in 1997. So, we subtract the initial year from the final year to get the difference. That would be 2000 - 1997 = 3. So there are 3 years between 1997 and 2000.

Step 3: Using linear growth formula

We can use the formula for a linear function, which looks like \(y = mx + b\). Here, 'm' is the rate of increase, 'x' is the time (here, number of years), and 'b' is the initial quantity (here, visitors in 1997). Here, 'm' is the rate of increase in visitors per year, which is 80,000. 'x' is the difference in a number of years, which we found out to be 3. 'b' is the number of visitors in 1997, represented by \(x\).

Step 4: Construct the expression

Now, just plug in the values into the equation. So we get \(y = 80000 * 3 + x\), which simplifies to \(y = 240000 + x\). Therefore, the number of visitors in 2000 can be represented by the expression \(240000 + x\).

Key Concepts

Rate of Change

The rate of change is a crucial concept when discussing how a quantity evolves over time. In the context of the given problem, the rate of change is exemplified by the increase in the number of American visitors to Ireland each year. Specifically, this rate is given as 80,000 visitors per year.

This rate of change represents the consistent annual increase in visitors, meaning each year 80,000 more visitors come to Ireland compared to the previous year. Understanding this concept helps us determine how much of a quantity, like visitor numbers, changes over a given period.

In a linear equation, the rate of change corresponds to the slope, often represented by the letter 'm' in the equation form \( y = mx + b \). This linear growth pattern ensures that regardless of the starting point, the change remains constant each year.

Expression

An expression is a mathematical phrase that combines numbers, variables, and operators to represent a particular value. In this exercise, we are tasked with writing an expression to calculate how many visitors came to Ireland in the year 2000, given the conditions specified.

We start with the number of visitors in 1997, which is denoted by the variable 'x'. Since we know there is a constant increase in visitors each year, we can represent the 2000 visitor number as an expression that adds the total increase over three years (3 years) to the 1997 number.

Mathematically, this expression is structured as \( y = 80,000 \times 3 + x \), which simplifies to \( y = 240,000 + x \). This expression effectively captures the visitor number in 2000 by incorporating both the initial amount and the increase over time.

Variable

A variable is a symbol or letter that represents a quantity in a mathematical expression or equation. In the provided exercise, the variable 'x' is used to denote the number of American visitors to Ireland in the base year, 1997. This base year provides a starting point, allowing us to track changes in visitor numbers over time.

Variables are integral to algebra as they enable flexibility and can be adjusted depending on the specific values that must be explored. In our expression, \( y = 240,000 + x \), the 'x' allows us to compute different outcomes depending on the initial number of visitors in 1997.

Understanding how to manipulate and use variables is key in constructing equations and expressions that accurately model real-world scenarios. By substituting different values for 'x', we can easily compute the number of visitors for various initial conditions.