Problem 57 - Algebra A Combined Function

Question

The sales \(y\) of VHS movie format units (in billions of dollars) sold in the United States from 2000 to 2008 is given by \(y=-1.1 x+7.1,\) where \(x\) is the number of years after 2000 . The sales \(y\) of DVD movie format units (in billions of dollars) sold in the United States from 2000 to 2008 is given by \(y=1.9 x+4.7,\) where \(x\) is the number of years after 2000. (Source: EMedia Digital Studio Magazine) a. Use the substitution method to solve this system of equations. $$ \left\\{\begin{array}{l} y=-1.1 x+7.1 \\ y=1.9 x+4.7 \end{array}\right. $$ Round \(x\) to the nearest tenth and \(y\) to the nearest whole number. b. Explain the meaning of your answer to part (a). c. Sketch a graph of the system of equations. Write a sentence describing the trends in the popularity of these two types of movie formats. d. Use the VHS equation to find the sales of VHS units in 2007. Then explain your answer.

Step-by-Step Solution

Answer

VHS and DVD sales were equal at about $6 billion in mid-2001. VHS sales were around $0.6 billion in 2007, showing a decline.

Step 1: Set Equations Equal

To solve the system of equations by substitution, set the two given equations equal to each other.\(-1.1x + 7.1 = 1.9x + 4.7\)

Step 2: Solve for x

Solve the equation from Step 1 to find the value of \(x\).Subtract \(1.9x\) from both sides:\(-1.1x - 1.9x + 7.1 = 4.7\)Combine the \(x\) terms:\(-3.0x + 7.1 = 4.7\)Subtract \(7.1\) from both sides:\(-3.0x = 4.7 - 7.1\)\(-3.0x = -2.4\)Divide by \(-3.0\):\(x = \frac{-2.4}{-3.0} = 0.8\)

Step 3: Substitute x to Find y

Now, substitute \(x = 0.8\) back into one of the original equations to find \(y\). Using the VHS equation:\(y = -1.1(0.8) + 7.1\)Calculate \(y\):\(y = -0.88 + 7.1\)\(y = 6.22\)Round \(y\) to the nearest whole number: \(y = 6\).

Step 4: Interpret the Solution

The solution \((x, y) = (0.8, 6)\) indicates that, approximately 0.8 years after 2000 (or around mid-2001), the sales of VHS and DVD formats in the US were the same, valued at approximately $6 billion.

Step 5: Graph the Equations

To sketch the graph, plot both equations on the same set of axes. The VHS line starts at (0, 7.1) with a negative slope of -1.1, while the DVD line starts at (0, 4.7) with a positive slope of 1.9. The intersection at about \(x = 0.8\) and \(y = 6\) confirms the point where VHS and DVD sales were equal. The graph shows VHS sales decreasing over time and DVD sales increasing.

Step 6: Find VHS Sales in 2007

For the year 2007, which is 7 years after 2000, substitute \(x = 7\) into the VHS equation:\(y = -1.1(7) + 7.1\)Calculate \(y\):\(y = -7.7 + 7.1\)\(y = -0.6\)VHS sales were approximately $0.6 billion in 2007, indicating a significant decline in sales over the period.

Step 7: Explain VHS Sales in 2007

The calculation shows that by 2007, VHS sales have significantly decreased to just $0.6 billion, reflecting the declining popularity of VHS as DVD sales rose.

Key Concepts

Substitution Method

The substitution method is a powerful tool used to solve systems of equations. It involves expressing one variable in terms of the other and then substituting this expression into the other equation. This reduces the system into a single equation with one variable, making it much easier to solve.
Let's break it down in simple steps:

• Start by solving one of the equations for one variable.
• Next, substitute this expression into the other equation. This creates a single equation that can be solved traditionally.
• After finding the value of one variable, use it to solve for the other variable by substituting back into any of the original equations.

In our current problem, setting the equations equal (\(-1.1x + 7.1 = 1.9x + 4.7\) ), we efficiently found the intersection point where both sales were equal. This method is especially effective for linear equations, offering a systematic approach to find solutions.

Linear Equations

Linear equations represent straight lines when graphed on a coordinate plane. Here, both VHS and DVD sales are described using linear equations, where each equation is in the form \(y = mx + b\). The term \(m\) is the slope, describing the steepness and direction of the line, and \(b\) is the y-intercept, showing where the line crosses the y-axis.
For the VHS sales, \(y = -1.1x + 7.1\), indicating a downward trend with a negative slope. This shows sales declining over time. On the other hand, the DVD sales equation \(y = 1.9x + 4.7\) has a positive slope, illustrating growth. These equations allow us to make predictions about sales over time and understand how trends shift in a market.
The key takeaway here is that linear equations help model real-world situations in a manageable and predictable way, allowing for analysis and forecasting based on patterns.

Intersection Point

The intersection point of two lines on a graph is where they cross, indicating that their equations are equal at that point. In the context of solving systems of equations, finding this point is crucial. It represents the solution to the system.
In our exercise, we found the intersection point to be \((x, y) = (0.8, 6)\).
This tells us that approximately 0.8 years after 2000, or around mid-2001, VHS and DVD movie format sales were equal at about $6 billion. To solve for the intersection, we used the substitution method discussed earlier, setting the equations equal and solving for the variables.
This observation is not only mathematically interesting but practically important, as it provides insights into shifting market dynamics over time.

Trend Analysis

Trend analysis involves studying patterns over time to predict future movements. In this case, we examined sales trends of VHS and DVD formats from 2000 to 2008 using their respective linear equations.
By analyzing these trends, we see that:

• The VHS sales decline steadily, implied by the negative slope of the line. This trend reflects diminishing consumer interest and ultimately, obsolescence.
• Conversely, DVD sales rise as indicated by the positive slope, showcasing growing consumer adoption.

Graphing these lines reveals much about the market transition from VHS to DVD.
The intersection, where sales were equal, further illuminates a crucial turning point in consumer preferences.
Understanding these trends is vital for businesses and analysts aiming to forecast changes and adjust strategies accordingly in related industries.