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.
Solve using the Elimination method
Part B
How long is each solo act?
Part A:
Let's represent the number of solo acts as "s" and the number of ensemble acts as "e".
According to the given information:
s + e = 14 (There are 12 solo acts and 2 ensemble acts in total)
Let's also represent the duration of each solo act as "x" and the duration of each ensemble act as "y".
According to the given information:
12x + 2y = 90 (The total duration of the show is 90 minutes)
Part B:
To solve using the Elimination method, we multiply the first equation by 2 to eliminate the "e" variable:
2s + 2e = 28
Now we have the following system of equations:
2s + 2e = 28
12x + 2y = 90
By subtracting the second equation from the first equation, we eliminate the "e" variable:
2s + 2e - (12x + 2y) = 28 - 90
2s - 12x = -62
To simplify, we divide the entire equation by 2:
s - 6x = -31
Now we have the following system of equations:
s - 6x = -31
12x + 2y = 90
To solve for x, let's isolate s in the first equation:
s = 6x - 31
Substituting this value of s into the second equation, we get:
12x + 2y = 90
12x + 2(6x - 31) = 90
12x + 12x - 62 = 90
24x = 152
x = 6.33
Therefore, each solo act lasts approximately 6.33 minutes.