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 x represent the duration of each solo act, and y represent the duration of each ensemble act.
We are given the following information:
1) The show will last 90 minutes: x(12) + y(2) = 90
(The duration of all solo acts is 12 times x, and the duration of all ensemble acts is 2 times y.)
2) The repeat show will last 60 minutes: x(6) + y(2) = 60
(The duration of the chosen solo acts is 6 times x, and the duration of the ensemble acts is 2 times y.)
Part B: From the equations above, we can solve for x:
x(12) + y(2) = 90
12x + 2y = 90
x(6) + y(2) = 60
6x + 2y = 60
To solve the equations, we can subtract the second equation from the first:
(12x + 2y) - (6x + 2y) = 90 - 60
6x = 30
x = 5
Therefore, each solo act lasts 5 minutes.