Your school's talent show will feature 14 solo acts and 3 ensemble acts. The show will last 114 minutes. The 7 solo performers judged best will give a repeat performance at a second 72 minute show, which will also feature the 3 ensemble acts. Each solo act lasts x minutes, and each ensemble act lasts y minutes. Use this information to answer parts (a) and (b).
Let the number of minutes each solo act lasts be x.
The 14 solo acts will therefore last a total of 14x minutes.
The 3 ensemble acts will last a total of 3y minutes.
The total duration of the talent show is 114 minutes, so we have the equation:
14x + 3y = 114.
In the second show, the 7 solo performers judged best will perform again. The duration of this show is 72 minutes, so we have the equation:
7x + 3y = 72.
We can now solve this system of equations to find the values of x and y.
To do this, we can multiply the second equation by 2 to eliminate y:
14x + 6y = 144.
Subtracting this equation from the first equation will eliminate x:
(14x + 3y) - (14x + 6y) = 114 - 144,
-3y = -30,
y = 10.
Substituting this value of y into the first equation, we can find x:
14x + 3(10) = 114,
14x + 30 = 114,
14x = 84,
x = 6.
Therefore, each solo act lasts 6 minutes and each ensemble act lasts 10 minutes.
(a) Each solo act lasts 6 minutes and each ensemble act lasts 10 minutes.
(b) The duration of the talent show is 114 minutes. The duration of the second show is 72 minutes.