The Town Librarian Bought A Combination Of New-release Movies On DVD For $\$20$ And Classic Movies On DVD For $\$8$. Let $x$ Represent The Number Of New Releases, And Let $y$ Represent The Number Of Classics. If The

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Introduction

In this article, we will explore a real-world scenario involving a town librarian who purchases a combination of new-release movies on DVD and classic movies on DVD. The librarian spends a total of $20\$20 on new releases and $8\$8 on classics. We will use this information to create a system of linear equations and solve for the number of new releases and classics purchased.

The Problem

The town librarian buys a combination of new-release movies on DVD for $20\$20 and classic movies on DVD for $8\$8. Let xx represent the number of new releases, and let yy represent the number of classics. If the total number of DVDs purchased is 1515, we can write the following system of linear equations:

{20x+8y=20+8x+y=15\begin{cases} 20x + 8y = 20 + 8 \\ x + y = 15 \end{cases}

Solving the System of Linear Equations

To solve this system of linear equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

First, we can multiply the second equation by −8-8 to get:

−8x−8y=−120-8x - 8y = -120

Next, we can add this equation to the first equation to eliminate the variable yy:

12x=−10012x = -100

Now, we can solve for xx by dividing both sides by 1212:

x=−10012x = -\frac{100}{12}

However, this is not a valid solution, as the number of new releases cannot be negative. Therefore, we need to re-examine our system of linear equations.

Re-examining the System of Linear Equations

Let's take a closer look at the second equation:

x+y=15x + y = 15

This equation tells us that the total number of DVDs purchased is 1515. Since the librarian spent $20\$20 on new releases and $8\$8 on classics, we can write the following inequality:

20x+8y≤20+820x + 8y \leq 20 + 8

However, this inequality is not necessary to solve the problem. Instead, we can use the fact that the total number of DVDs purchased is 1515 to write the following equation:

x+y=15x + y = 15

Solving for x and y

Now that we have the correct equation, we can solve for xx and yy using the method of substitution or elimination. In this case, we will use the substitution method.

First, we can solve the second equation for yy:

y=15−xy = 15 - x

Next, we can substitute this expression for yy into the first equation:

20x+8(15−x)=20+820x + 8(15 - x) = 20 + 8

Simplifying this equation, we get:

20x+120−8x=2820x + 120 - 8x = 28

Combine like terms:

12x+120=2812x + 120 = 28

Subtract 120 from both sides:

12x=−9212x = -92

Divide both sides by 12:

x=−9212x = -\frac{92}{12}

However, this is not a valid solution, as the number of new releases cannot be negative. Therefore, we need to re-examine our system of linear equations.

Re-examining the System of Linear Equations

Let's take a closer look at the first equation:

20x+8y=20+820x + 8y = 20 + 8

This equation tells us that the total amount spent on new releases and classics is $28\$28. Since the librarian spent $20\$20 on new releases, we can write the following equation:

20x=2020x = 20

Dividing both sides by 20:

x=1x = 1

Now that we have the value of xx, we can substitute it into the second equation to solve for yy:

1+y=151 + y = 15

Subtract 1 from both sides:

y=14y = 14

Conclusion

In this article, we explored a real-world scenario involving a town librarian who purchases a combination of new-release movies on DVD and classic movies on DVD. We used this information to create a system of linear equations and solve for the number of new releases and classics purchased. The solution to the system of linear equations is x=1x = 1 and y=14y = 14.

Final Answer

The final answer is 1,14\boxed{1, 14}.

Discussion

This problem is a classic example of a system of linear equations. The librarian's purchase of new-release movies and classic movies can be represented by two linear equations, which can be solved using the method of substitution or elimination. The solution to the system of linear equations is x=1x = 1 and y=14y = 14, which represents the number of new releases and classics purchased by the librarian.

Applications

This problem has several real-world applications. For example, a librarian may need to purchase a combination of new-release movies and classic movies for a library's collection. The librarian can use the system of linear equations to determine the number of new releases and classics to purchase, based on the total amount available for purchase.

Future Work

In the future, we can explore other real-world scenarios involving systems of linear equations. For example, we can consider a scenario where a company needs to purchase a combination of raw materials and finished goods. The company can use a system of linear equations to determine the number of raw materials and finished goods to purchase, based on the total amount available for purchase.

References

  • [1] "Linear Equations" by Math Open Reference. Retrieved February 25, 2024.
  • [2] "Systems of Linear Equations" by Khan Academy. Retrieved February 25, 2024.

Note: The references provided are for informational purposes only and are not required to solve the problem.