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?
explain using the elimination method
Part A:
Let's introduce variables:
x = duration of each solo act in minutes
y = duration of each ensemble act in minutes
The total number of solo acts is 12, and the total number of ensemble acts is 2. Each solo act and ensemble act lasts a certain duration, so the total duration of the solo acts can be represented as 12x, and the total duration of the ensemble acts can be represented as 2y.
The total duration of the first show is 90 minutes, so we can set up the equation:
12x + 2y = 90
The total duration of the second show is 60 minutes, and it only includes the 6 best solo acts (which will be repeated) and the 2 ensemble acts. Therefore, the total duration of the solo acts in the second show is 6x, and the total duration of the ensemble acts remains the same at 2y.
The equation for the second show is:
6x + 2y = 60
Therefore, the system of equations to model the situation is:
12x + 2y = 90
6x + 2y = 60
Part B:
To solve for the duration of each solo act, we can use the elimination method. We'll eliminate the variable y by subtracting the second equation from the first equation.
(12x + 2y) - (6x + 2y) = 90 - 60
6x = 30
x = 30/6
x = 5
Therefore, each solo act lasts for 5 minutes.