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 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. (type an ordered - pair using integers or decimals.) in 2018, the revenue from sales of cds was the same as the revenue from sales of vinyl records. c. sketch a graph of the system of equations. choose the correct graph below. a. graph image b. graph image c. graph image d. graph image write a sentence describing the trends for revenue from sales of cds and vinyl records. before the intersection point, the revenue from sales of cds was blank. after the intersection, the revenue from sales of vinyl records is blank.

Explanation:

Step1: Analyze the equations

We have two - linear equations for revenue. For CDs, $y=-235x + 1853.5$ and for vinyl records, $y = 44x+458.5$, where $x$ is the number of years after 2013.

Step2: Interpret the solution of the system

The solution of the system $(5,678.5)$ means that 5 years after 2013 (i.e., in 2018), the revenue from sales of CDs and the revenue from sales of vinyl records are both $678.5$ dollars.

Step3: Analyze the graphs

To sketch the graph of the system of equations, we know that the equation for CDs $y=-235x + 1853.5$ has a $y$ - intercept of $1853.5$ and a slope of $-235$ (a decreasing line). The equation for vinyl records $y = 44x+458.5$ has a $y$ - intercept of $458.5$ and a slope of $44$ (an increasing line). The two lines will intersect at the point $(5,678.5)$.

Step4: Describe the trends

Before the intersection point (before 2018), the revenue from sales of CDs is higher than the revenue from sales of vinyl records. After the intersection point (after 2018), the revenue from sales of vinyl records is higher than the revenue from sales of CDs.

Answer:

a. In 2018 (5 years after 2013), the revenue from sales of CDs and the revenue from sales of vinyl records are both $678.5$ dollars.
b. The solution $(5,678.5)$ of the system means that 5 years after 2013, the revenue from CD sales and vinyl - record sales is the same, which is $678.5$ dollars.
c. To sketch the graph:

• For the CD - revenue equation $y=-235x + 1853.5$, the $y$ - intercept is $(0,1853.5)$ and using the slope of $-235$, we can find other points.
• For the vinyl - record revenue equation $y = 44x+458.5$, the $y$ - intercept is $(0,458.5)$ and using the slope of $44$, we can find other points. The two lines intersect at $(5,678.5)$.

Before the intersection point, the revenue from sales of CDs is higher than that of vinyl records. After the intersection point, the revenue from sales of vinyl records is higher than that of CDs.