Jarred sells DVDs. His inventory shows that he has a total of 3,500 DVDs. He has 2,342 more contemporary titles than classic titles. Let [tex]$x$[/tex] represent the number of contemporary titles and [tex]$y$[/tex] represent the number of classic titles. The system of equations models the given information:
[tex]
\begin{array}{l}
x+y=3,500 \\
x-y=2,342
\end{array}
[/tex]
Solve the system of equations. How many contemporary titles does Jarred have?
Answer (1)
Set up the system of equations: x + y = 3500 and x − y = 2342 .
Add the two equations to eliminate y : 2 x = 5842 .
Divide by 2 to solve for x : x = 2921 .
State the number of contemporary titles: 2921 .
Explanation
Understanding the Problem We are given a system of two equations with two variables, x and y , representing the number of contemporary and classic DVD titles, respectively. The equations are:
x + y = 3500 x − y = 2342
Our goal is to find the value of x , which represents the number of contemporary titles.
Eliminating y To solve this system of equations, we can use the elimination method. We add the two equations together to eliminate the variable y :
( x + y ) + ( x − y ) = 3500 + 2342
Simplifying the left side, we get:
2 x = 3500 + 2342
Adding the Equations Now, we add the numbers on the right side:
2 x = 5842
Solving for x To find the value of x , we divide both sides of the equation by 2:
x = 2 5842
x = 2921
Finding the Answer Therefore, the number of contemporary titles is 2921.
Examples
Imagine you're managing a store's inventory, just like Jarred with his DVDs. Knowing how to solve a system of equations helps you determine the quantities of different items you have in stock, especially when you have relationships between those quantities. For example, if you know the total number of items and the difference between two types of items, you can use a system of equations to find the exact quantity of each type. This is useful for ordering, tracking sales, and making informed business decisions. In this case, we found that Jarred has 2921 contemporary DVDs, which helps him manage his inventory effectively.
