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. 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. 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 revenue from sales of cds was the revenue from sales of vinyl records.

Explanation:

Step1: Recall substitution - method concept

The system of equations is \(y = - 235x+1853.5\) (for CDs) and \(y = 44x + 458.5\) (for vinyl records). In the substitution method, we set the two right - hand sides equal to each other since \(y\) is the same in both equations: \(-235x + 1853.5=44x + 458.5\).

Step2: Solve for \(x\)

First, add \(235x\) to both sides: \(1853.5=44x+235x + 458.5\). Then combine like terms: \(1853.5 = 279x+458.5\). Next, subtract \(458.5\) from both sides: \(1853.5 - 458.5=279x\), so \(1395 = 279x\). Divide both sides by \(279\) to get \(x = 5\).

Step3: Solve for \(y\)

Substitute \(x = 5\) into the equation \(y = 44x+458.5\). Then \(y=44\times5 + 458.5=220+458.5 = 678.5\).

Step4: Explain the meaning of the solution

The \(x\) - value represents the number of years after 2013. So \(x = 5\) means 5 years after 2013, which is 2018. The \(y\) - value represents the revenue in millions of dollars. The revenue from CD sales and vinyl record sales is the same (\(y = 678.5\) million dollars) in the year 2018.

Step5: 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 \((5,678.5)\) on the graph.

Answer:

a. We set \(-235x + 1853.5=44x + 458.5\), combined like terms, solved for \(x\) to get \(x = 5\), and then substituted \(x = 5\) into \(y = 44x+458.5\) to get \(y = 678.5\).
b. The solution \((5,678.5)\) means that 5 years after 2013 (in 2018), the revenue from sales of CDs and vinyl records is the same, which is 678.5 million dollars.
c. The graph of \(y=-235x + 1853.5\) is a line with a negative slope and \(y\) - intercept of \(1853.5\), and the graph of \(y = 44x+458.5\) is a line with a positive slope and \(y\) - intercept of \(458.5\). They intersect at the point \((5,678.5)\).