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 revenues (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.

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.

Explanation:

Step1: Identify the equations

The equations for CD - sales revenue is $y=-235x + 1853.5$ and for vinyl - record sales revenue is $y = 44x+458.5$, where $x$ is the number of years after 2013.

Step2: Use the substitution method

Since both equations are equal to $y$, we can set them equal to each other:
$-235x + 1853.5=44x+458.5$.
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 $1395 = 279x$.
Divide both sides by 279: $x=\frac{1395}{279}=5$.
Substitute $x = 5$ into $y = 44x+458.5$: $y=44\times5+458.5=220 + 458.5=678.5$.

Step3: Explain the meaning of the solution

The ordered pair $(5,678.5)$ 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 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 will intersect at the point $(5,678.5)$.

Answer:

a. The system of equations is

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

, where $x$ is the number of years after 2013. The solution using the substitution method:
Set $-235x + 1853.5=44x+458.5$.
$1853.5-458.5=44x + 235x$.
$1395=279x$, so $x = 5$.
Substitute $x = 5$ into $y = 44x+458.5$, $y=44\times5+458.5=678.5$. The solution is the ordered pair $(5,678.5)$.
b. The ordered pair $(5,678.5)$ means that in 2018 (5 years after 2013), the revenue from sales of CDs and the revenue from sales of vinyl records are both 678.5 million dollars.
c. To sketch the graph:
For $y=-235x + 1853.5$, the $y$ - intercept is 1853.5 and the slope is - 235.
For $y = 44x+458.5$, the $y$ - intercept is 458.5 and the slope is 44. The two lines intersect at the point $(5,678.5)$.