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 = - 236x + 1953.5 ), while for vinyl records, the equation is ( y = 42x + 285.5 ), 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 (square).

(type an ordered pair using integers or decimals.)

b. explain the meaning of the solution of the system.

in (square) the revenue from sales of cds was (square) the revenue from sales of vinyl records.

c. sketch a graph of the system of equations. choose the correct graph below.

o a.

o b.

o c.

o d.

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 (square), the revenue from sales of vinyl records. after

Explanation:

Step1: Substitute the equations

We have the system of equations \(y = - 236x+1953.5\) (for CDs) and \(y = 42x + 285.5\) (for vinyl records). Since both equal \(y\), we can set them equal to each other: \(-236x + 1953.5=42x+285.5\).

Step2: Solve for \(x\)

Add \(236x\) to both sides: \(1953.5=42x + 236x+285.5\), which simplifies to \(1953.5 = 278x+285.5\).
Subtract \(285.5\) from both sides: \(1953.5 - 285.5=278x\), so \(1668 = 278x\).
Divide both sides by \(278\): \(x=\frac{1668}{278}=6\).

Step3: Solve for \(y\)

Substitute \(x = 6\) into \(y = 42x+285.5\). Then \(y=42\times6 + 285.5=252+285.5 = 537.5\).

Answer:

The solution of the system is \((6,537.5)\)