Question

in the music industry, revenue from sales of cds has been declining in recent years, while the revenue from sales of vinyl records has been increasing. for the years 2013 through 2020, the revenue (in millions of dollars) from the sale of cds is approximated by the equation y = - 235x + 1853.5, while for vinyl records, the equation is y = 44x + 458.5, where x is the number of years after 2013. a. use the substitution method to solve this system of equations. b. explain the meaning of the solution of the system. c. sketch a graph of the system of equations. write a sentence describing the trends for revenue from sales of cds and vinyl records. the solution of the system is (5,678.5) (type an ordered - pair using integers or decimals.) b. explain the meaning of the solution of the system. in the revenue from sales of cds was the revenue from sales of vinyl records

Explanation:

Step1: Write the system of equations

Let \(y_1=- 235x + 1853.5\) (revenue from CDs) and \(y_2 = 44x+458.5\) (revenue from vinyl records). The system of equations is \(

$$\begin{cases}y=-235x + 1853.5\\y = 44x+458.5\end{cases}$$

\)

Step2: Use substitution method

Since both expressions equal \(y\), we set \(-235x + 1853.5=44x+458.5\).
Add \(235x\) to both sides: \(1853.5=44x + 235x+458.5\).
Combine like - terms: \(1853.5 = 279x+458.5\).
Subtract \(458.5\) from both sides: \(1853.5−458.5=279x\), so \(1395 = 279x\).
Solve for \(x\): \(x=\frac{1395}{279}=5\).
Substitute \(x = 5\) into \(y = 44x+458.5\), then \(y=44\times5+458.5=220 + 458.5=678.5\).

Step3: Explain the meaning of the solution

The \(x - value\) of \(5\) represents the number of years after 2013, so the year is \(2013 + 5=2018\). The \(y - value\) of \(678.5\) represents the revenue (in millions of dollars), and at \(x = 5\) (year 2018), the revenue from sales of CDs and vinyl records is the same, which is \(678.5\) million dollars.

Step4: Sketch the graph

The equation \(y=-235x + 1853.5\) has a \(y\) - intercept of \(1853.5\) and a slope of \(-235\). The equation \(y = 44x+458.5\) has a \(y\) - intercept of \(458.5\) and a slope of \(44\). The two lines intersect at the point \((5,678.5)\).

Answer:

a. The system of equations is \(

$$\begin{cases}y=-235x + 1853.5\\y = 44x+458.5\end{cases}$$

\). Using substitution:

• Set \(-235x + 1853.5=44x+458.5\).
• Solve for \(x\): \(x = 5\).
• Substitute \(x = 5\) into \(y = 44x+458.5\) to get \(y=678.5\).

b. The \(x = 5\) means 5 years after 2013 (the year 2018), and the \(y = 678.5\) means the revenue from sales of CDs and vinyl records is \(678.5\) million dollars in the year 2018.
c. Sketch two lines \(y=-235x + 1853.5\) and \(y = 44x+458.5\) which intersect at the point \((5,678.5)\).