Currently, Jodi has a collection of 44 CDs and Veronica has a collection of 22 CDs. Jodi is adding 2 CDs per month to her collection; Veronica plans to add 4 CDs per month.
a. Write a system of equations for this situation. Explain your method.
b. Graph the system of equations. Describe your graph.
c. In how many months will they have the same number of CDs? What will be the number of CDs? Explain how you found the answer.
d. At the end of 2 months, who will have more CDs? Explain how you found the answer.
J=44+2T
V=22+4T
a. To write a system of equations for this situation, we need to represent the number of CDs each person has with respect to the number of months.
Let's assume t represents the number of months that have passed.
For Jodi: Since she is adding 2 CDs per month to her collection, her equation would be:
Jodi(t) = 44 + 2t
For Veronica: Since she is adding 4 CDs per month to her collection, her equation would be:
Veronica(t) = 22 + 4t
b. To graph the system of equations, we will plot Jodi's and Veronica's equations on a coordinate plane, using the number of months (t) on the x-axis and the number of CDs on the y-axis. The graph will show how the number of CDs for each person changes over time.
c. To find the number of months when they will have the same number of CDs, we need to set Jodi(t) equal to Veronica(t) and solve for t.
44 + 2t = 22 + 4t
First, let's subtract 44 from both sides:
2t = 22 + 4t - 44
2t = -22 + 4t
Next, let's subtract 4t from both sides:
2t - 4t = -22
-2t = -22
To isolate t, we'll divide both sides by -2:
t = 11
Therefore, they will have the same number of CDs after 11 months.
To find the number of CDs at that time, we can substitute t = 11 into either Jodi(t) or Veronica(t).
Jodi(11) = 44 + 2(11) = 44 + 22 = 66 CDs.
Therefore, after 11 months, they will both have 66 CDs each.
d. To determine who will have more CDs after 2 months, we can compare the values of Jodi(2) and Veronica(2).
Jodi(2) = 44 + 2(2) = 44 + 4 = 48 CDs.
Veronica(2) = 22 + 4(2) = 22 + 8 = 30 CDs.
Therefore, after 2 months, Jodi will have more CDs with a total of 48 CDs, whereas Veronica will have 30 CDs.