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?
work using the elimination method.
Part A:
Let's assume that each solo act lasts x minutes and each ensemble act lasts y minutes.
Since there will be 12 solo acts and 2 ensemble acts in the first show, the total number of acts in the first show is 12 + 2 = 14.
The total time for the first show is given as 90 minutes.
So, the first equation based on the total time of the first show is:
12x + 2y = 90
In the second show, only the 6 best solo acts are repeated. So, the total number of acts in the second show is 6 + 2 = 8.
The total time for the second show is given as 60 minutes.
So, the second equation based on the total time of the second show is:
6x + 2y = 60
Part B:
To solve this system of equations using the elimination method, we can multiply the first equation by 3 and the second equation by 5 to eliminate the y term:
First equation: 3(12x + 2y) = 3(90)
36x + 6y = 270
Second equation: 5(6x + 2y) = 5(60)
30x + 10y = 300
Now, subtract the second equation from the first equation:
(36x + 6y) - (30x + 10y) = 270 - 300
6x - 4y = -30
Since the coefficients of the y term are the same, subtracting the two equations eliminated the y term, leaving us with a single equation in terms of x.
To solve for x, divide both sides of the equation by 6:
(6x - 4y)/6 = -30/6
x - (2/3)y = -5
So, the equation in terms of x is x - (2/3)y = -5.
Since we're asked to find the length of each solo act, we're looking for the value of x. However, the equation in terms of x and y doesn't provide enough information to determine the exact value of x. To find the length of each solo act, we would need additional information or equations.