Your school's talent show will feature 16 solo acts and 3 ensemble acts. The show will last 129 minutes. The 8 solo performers judged best will give a repeat performance at a second 81 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).
To answer parts (a) and (b), we need to use the given information about the number of acts and the duration of the shows.
Let's start by assigning variables to the duration of solo acts and ensemble acts. Let's say each solo act lasts x minutes and each ensemble act lasts y minutes.
(a) We know that there will be 16 solo acts and 3 ensemble acts. So the total duration of the solo acts in the first show is 16x minutes, and the total duration of the ensemble acts in the first show is 3y minutes. The first show lasts 129 minutes in total. Therefore, we can write the equation:
16x + 3y = 129
(b) We also know that the best 8 solo performers will give a repeat performance in the second show. The second show also includes the 3 ensemble acts. So the total duration of the solo acts in the second show is 8x minutes, and the total duration of the ensemble acts in the second show is 3y minutes. The second show lasts 81 minutes in total. Therefore, we can write the equation:
8x + 3y = 81
Now, we have a system of two equations with two variables (x and y). We can solve this system of equations to find the values of x and y.
b) Solve the system from part (a)
[ ] (Type an ordered pair.)
To solve the system of equations from part (a), we can use the method of substitution or elimination.
Let's use the elimination method to solve the system.
Multiplying the first equation by 8 and the second equation by 16, we get:
128x + 24y = 8(129) (equation 1)
128x + 48y = 16(81) (equation 2)
Subtracting equation 1 from equation 2, we get:
(128x + 48y) - (128x + 24y) = 16(81) - 8(129)
24y - 24y = 1296 - 1032
0 = 264
This yields a contradiction. It means that there is no solution to the system of equations.
Therefore, there is no ordered pair (x, y) that satisfies both equations simultaneously.