Mrs. Todd is a dance teacher whose students are going to have a recital. She is comparing two companies' prices for filming the recital and producing DVDs. Belle's Productions charges $2 per DVD, plus $80 to come and film the recital. Oak Grove Media charges $70 to film, plus $4 per DVD. To decide which company to use, Mrs. Todd determines the number of DVDs that would make the two options equivalent in terms of cost. How many DVDs would that be? Write a system of equations, graph them, and type the solution.
Let's represent the number of DVDs by "x".
For Belle's Productions:
Cost of filming = $80
Cost of DVDs = $2x
Total Cost for Belle's Productions = $80 + $2x
For Oak Grove Media:
Cost of filming = $70
Cost of DVDs = $4x
Total Cost for Oak Grove Media = $70 + $4x
To find the number of DVDs that would make the two options equivalent in terms of cost, we can set the two equations equal to each other:
$80 + $2x = $70 + $4x
Simplifying the equation:
$2x - $4x = $70 - $80
-$2x = -$10
Dividing both sides by -2:
x = 5
Therefore, the two options would be equivalent in terms of cost if Mrs. Todd produces 5 DVDs.
Let's assume the number of DVDs that would make the two options equivalent in terms of cost is represented by the variable "x".
For Belle's Productions, the cost is $2 per DVD plus $80 to come and film the recital. So the total cost for Belle's Productions can be represented by the equation:
Cost = 2x + 80
For Oak Grove Media, the cost is $70 to film plus $4 per DVD. So the total cost for Oak Grove Media can be represented by the equation:
Cost = 4x + 70
To find the number of DVDs that would make the two options equivalent in terms of cost, we set the two equations equal to each other:
2x + 80 = 4x + 70
Subtracting 2x from both sides:
80 = 2x + 70
Subtracting 70 from both sides:
10 = 2x
Dividing both sides by 2:
5 = x
Therefore, the number of DVDs that would make the two options equivalent in terms of cost is 5.
To graph the equations, we can plot the cost for each company in terms of the number of DVDs and find their intersection point. However, this solution has been already calculated and the number of DVDs is given as 5.
To solve this problem, we can start by setting up a system of equations. Let's assume that "x" represents the number of DVDs.
For Belle's Productions, the total cost is given by the equation:
Cost1 = 2x + 80
For Oak Grove Media, the total cost is given by the equation:
Cost2 = 70 + 4x
To find the number of DVDs that would make the two options equivalent in terms of cost, we set Cost1 equal to Cost2 and solve for "x":
2x + 80 = 70 + 4x
Now, we can solve this equation to find the value of "x".
2x - 4x = 70 - 80
-2x = -10
Dividing both sides by -2 gives:
x = (-10) / (-2)
x = 5
Therefore, the two options would be equivalent in cost when 5 DVDs are produced.
To graphically represent the solution, we can plot the cost for each option against the number of DVDs on a graph. The x-axis represents the number of DVDs, and the y-axis represents the cost.
For Belle's Productions, the equation is Cost1 = 2x + 80.
For Oak Grove Media, the equation is Cost2 = 70 + 4x.
Plotting these equations on the graph, we can see where they intersect, which will indicate the number of DVDs that would make the two options equivalent in terms of cost.
The solution is x = 5, which means that 5 DVDs would make the two options equivalent in terms of cost.