Problem 23 - Algebra 1

Question

Use the table which shows the number of dollars (in billions) spent on books and maps in the United States from 1990 through 1995. $$ \begin{array}{|l|c|c|c|c|c|c|}\hline \text { Years since 1990 } & 0 & 1 & 2 & 3 & 4 & 5 \\\\\hline \text { Billions of dollars } & 16.5 & 16.9 & 17.7 & 19.0 & 20.1 & 20.9 \\\\\hline\end{array} $$ Write a linear model for the amount spent on books and maps.

Step-by-Step Solution

Answer

The linear model representing the cost of books and maps from 1990 through 1995 is \(y = 0.88x + 16.5\).

Step 1: Identify the Change

the change in the number of billions of dollars spent from year to year is constant. Most of the time, the amount spent increases. This change can be identified by subtracting the amount spent in the previous year from the amount spent in the current year. For instance, the change from 1990 to 1991 is \(16.9 - 16.5 = 0.4\) billion dollars. Similarly, we can find the changes for the rest of the years.

Step 2: Calculate the Average Rate of Change

The average rate of change over a period of time, in this case 'years since 1990', is equivalent to the slope of the linear model we are trying to create. It can be calculated by dividing the total change in the amount spent (the difference between the amount spent in 1995 and 1990) by total number of years. Therefore, the average rate of change is \((20.9 - 16.5)/(5) =0.88\) billion dollars.

Step 3: Write the Linear model

A linear model can be written in the slope-intercept form \(y = mx + b\), where 'm' is the slope and 'b' is the y-intercept. In this context, 'y' is the billions of dollars spent, 'x' is the years since 1990, 'm' is the average rate of change, and 'b' is the initial amount spent in 1990. Therefore, the linear model would be written as \(y = 0.88x + 16.5\).

Key Concepts

Slope

When we talk about the slope in the context of linear equations, we refer to the steepness or incline of a line on a graph. The slope is an essential part of understanding linear relationships. It shows how much one quantity changes relative to another.

• In our example, we look at how the spending on books and maps changes year by year.
• The slope is calculated as the change in the y-values over the change in the x-values.

The slope is symbolized by 'm' and can be positive or negative. A positive slope indicates an increase, whereas a negative slope indicates a decrease. To find the slope in our exercise, we calculate it by taking the difference in the number of billions of dollars for each year and dividing by the change in years. This gives us a constant value, showing a steady increase in spending over time.

Rate of Change

The rate of change is essentially the slope of a linear equation. It tells us how much one variable changes in relation to another.

• In the textbooks exercise, the rate of change shows how the book and map expenses change over years.
• The rate of change here is $0.88 billion dollars per year.

Calculating the rate of change involves dividing the total change in the amount spent (from the first year to the last) by the number of years that have passed. This gives a positive figure, meaning an upward trend. Understanding the rate of change helps us predict future values based on past data, which is central to creating a linear model.

Slope-Intercept Form

The slope-intercept form of a linear equation is a way of writing equations of lines so that you can readily identify the slope and y-intercept. The general form of the equation is \( y = mx + b \).

• 'm' is the slope, which indicates the rate of change.
• 'b' is the y-intercept, which represents the initial value when x is zero.

In our context:

• 'y' is the billions spent on books and maps.
• 'x' represents the years since 1990.

For our problem, we already calculated the slope \( m = 0.88 \) and identified the y-intercept as the starting expenditure (16.5 billion dollars in 1990). Hence, the slope-intercept form becomes \( y = 0.88x + 16.5 \). This equation allows us to predict future spending or understand past trends by inputting different values of 'x'.