A snack bar sells two sizes of snack packs. A large snack pack is $5, and a small snack pack is $3. In one day, the snack bar sold 60 snack packs for a total of $220.
Part A
Write a system of equations that represents the scenario above. Let x = small snack packs and y = large snack packs.
Part B
How many small snack packs did the snack bar sell? Use the Elimination Method to solve. You must show work.
Part A:
The total number of small snack packs sold is x, and the total number of large snack packs sold is y.
The cost of each small snack pack is $3, so the equation for the cost of small snack packs is 3x.
The cost of each large snack pack is $5, so the equation for the cost of large snack packs is 5y.
The total cost of all the snack packs sold is $220, so the equation for the total cost is 3x + 5y = 220.
Therefore, the system of equations is:
3x + 5y = 220
Part B:
To solve this system of equations using the elimination method, we can multiply the first equation by -3 to eliminate the x term:
-3(3x + 5y) = -3(220)
-9x - 15y = -660
Now we can add this equation to the second equation to eliminate the x term:
-9x - 15y + 3x + 5y = -660 + 0
-6x - 10y = -660
Divide this equation by -2 to simplify:
-6x/-2 - 10y/-2 = -660/-2
3x + 5y = 330
Now we have a new system of equations:
3x + 5y = 330
3x + 5y = 220
If we subtract the second equation from the first equation, we can eliminate the y term:
(3x + 5y) - (3x + 5y) = 330 - 220
0 = 110
Since 0 does not equal 110, this system of equations has no solution.
Therefore, it is not possible to determine the number of small snack packs that the snack bar sold using the elimination method.
Your school's talent show will feature 12 solo acts and 2 ensemble acts. The show will last 90 minutes. The 6 solo performers judged best will give a repeat performance at a second 60-minute show, which will also feature the 2 ensemble acts. Each solo act lasts x minutes, and each ensemble act lasts y minutes.
Part A
Write a system of equations to model the situation.
Part B
How long is each solo act?
Part A:
Let's define the variables:
x = duration of each solo act (in minutes)
y = duration of each ensemble act (in minutes)
There are 12 solo acts, and each solo act lasts x minutes. So the total duration of the solo acts is 12x.
There are 2 ensemble acts, and each ensemble act lasts y minutes. So the total duration of the ensemble acts is 2y.
The first show lasts 90 minutes. It features the 12 solo acts (12x) and the 2 ensemble acts (2y). So the total duration of the first show is 12x + 2y = 90.
The second show lasts 60 minutes. It features the 6 solo acts (6x) judged best from the first show, and the 2 ensemble acts (2y). So the total duration of the second show is 6x + 2y = 60.
Therefore, the system of equations that models the situation is:
12x + 2y = 90
6x + 2y = 60
Part B:
We can solve the system of equations to find the duration of each solo act (x). By subtracting the second equation from the first equation, we can eliminate the y term:
(12x + 2y) - (6x + 2y) = 90 - 60
6x = 30
Divide both sides of the equation by 6:
x = 5
Therefore, each solo act lasts 5 minutes.