Problem 57 - Algebra A Combined Function

Question

Assume each exercise describes a linear relationship. Write the equations in slope-intercept form. In \(2003,\) there were 302 million magazine subscriptions in the United States. By 2007 , this number was 322 million. (Source: Audit Bureau of Circulation, Magazine Publishers Association) a. Write two ordered pairs of the form (years after \(2003,\) millions of magazine subscriptions) for this situation. b. Assume the relationship between years after 2003 and millions of magazine subscriptions is linear over this period. Use the ordered pairs from part (a) to write an equation for the line relating year after 2003 to millions of magazine subscriptions. C. Use this linear equation in part (b) to estimate the millions of magazine subscriptions in 2005 .

Step-by-Step Solution

Answer

In 2005, there were approximately 312 million magazine subscriptions.

Step 1: Define Ordered Pairs

First, identify the given points in the form of ordered pairs wherein the x-coordinate represents the number of years after 2003, and the y-coordinate represents the millions of magazine subscriptions. Therefore, the year 2003 corresponds to zero years after 2003 and the year 2007 corresponds to four years after 2003. So the ordered pairs are (0, 302) and (4, 322).

Step 2: Find the Slope

To find the slope of the line, we use the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Substituting the values from our ordered pairs, \( (x_1, y_1) = (0, 302) \) and \( (x_2, y_2) = (4, 322) \), we calculate: \[ m = \frac{322 - 302}{4 - 0} = \frac{20}{4} = 5 \]. Thus, the slope is 5.

Step 3: Write the Equation in Slope-Intercept Form

The slope-intercept form of a linear equation is \( y = mx + b \). We already have \( m = 5 \) from the slope calculation. Using the y-intercept which is the value of y when x = 0, which is 302 from the point (0, 302), the equation becomes \( y = 5x + 302 \).

Step 4: Estimate for the Year 2005

To estimate the millions of magazine subscriptions in 2005, we need to substitute \( x = 2 \) (as 2005 is 2 years after 2003) into the equation \( y = 5x + 302 \). Calculating this gives: \( y = 5(2) + 302 = 10 + 302 = 312 \).

Key Concepts

slope-intercept form

The slope-intercept form is a way of writing linear equations using the equation format: \(y = mx + b\). This formula provides a clear method to express a line equation by emphasizing two crucial components: the slope (m) and the y-intercept (b). The slope represents the rate of change of y concerning x, which means it describes how much y will increase or decrease as x increases by one unit. The y-intercept (b) indicates the point where the line crosses the y-axis, giving the value of y when x equals zero.
When you have a linear situation, such as counting magazine subscriptions over the years, using the slope-intercept form helps to represent this relationship as a straight line.
This form is incredibly helpful because it allows students and others to easily understand and predict changes by simply plugging different x values into the formula to find corresponding y values.

ordered pairs

Ordered pairs are a fundamental concept when dealing with points on a graph. They are written in the form \((x, y)\), where 'x' is the horizontal value indicating the position on the x-axis, and 'y' is the vertical value indicating the position on the y-axis.
In the exercise example, ordered pairs like \((0, 302)\) and \((4, 322)\) tell us the relationship between the years since 2003 and the number of magazine subscriptions. Here, '0' and '4' represent years after 2003, and '302' and '322' show millions of magazine subscriptions in those respective years.
Ordered pairs are used as points that help plot the line on a graph, and they give us specific data that allows for constructing linear equations and analyzing trends.

slope calculation

Calculating the slope is a key step in understanding how one variable affects another in linear relationships. The slope, denoted as \(m\), is calculated using the formula: \[m = \frac{y_2 - y_1}{x_2 - x_1}\],where \((x_1, y_1)\) and \((x_2, y_2)\) are two points on the line.
This calculation tells us the average rate at which the dependent variable \(y\) changes concerning the independent variable \(x\). A positive slope indicates a positive relationship, where as x increases, y also increases. Conversely, a negative slope indicates a negative relationship.
In our example, using the ordered pairs \((0, 302)\) and \((4, 322)\), the slope is:\[m = \frac{322 - 302}{4 - 0} = \frac{20}{4} = 5\].
This means that for each additional year after 2003, the millions of magazine subscriptions increased by 5. Understanding slope is crucial for predicting future values and understanding the nature of the linear relationship in a graph.