Your school's talent show will feature 14 solo acts and 2 ensemble acts. The show will last 110 minutes. The 7 solo performers judged best will give a repeat performance at a second 68 minute show, which will also feature the 2 ensemble acts. Each solo act lasts x minutes, and each ensemble act lasts y minutes. Use this information to answer parts (a) and (b).
(a) Let's assume that each solo act lasts for x minutes.
Since there are 14 solo acts, the total time taken by solo acts can be calculated as 14x minutes.
We also know that there are 2 ensemble acts, and each ensemble act lasts y minutes. Therefore, the total time taken by ensemble acts can be calculated as 2y minutes.
The total time taken by solo and ensemble acts is given as 110 minutes. Hence, we can write the equation as:
14x + 2y = 110
(b) In the second show, 7 solo performers judged best will give a repeat performance and the show will last for 68 minutes.
Hence, the time taken by the repeat performance of the 7 solo acts can be calculated as 7x minutes.
The total time taken by solo and ensemble acts in the second show is given as 68 minutes. Hence, we can write the equation as:
7x + 2y = 68
b) Solve the system from part (a).
enter your response here (Type an ordered pair.)
To solve the system of equations:
14x + 2y = 110
7x + 2y = 68
We can solve it using the following steps:
1. Subtract the second equation from the first equation to eliminate y:
(14x + 2y) - (7x + 2y) = 110 - 68
14x - 7x + 2y - 2y = 42
7x = 42
x = 42/7
x = 6
2. Substitute the value of x into either of the original equations to solve for y. Using the first equation:
14(6) + 2y = 110
84 + 2y = 110
2y = 110 - 84
2y = 26
y = 26/2
y = 13
Therefore, the solution to the system of equations is (x, y) = (6, 13).
How long is each solo act?
Each solo act lasts for 6 minutes.
How long is each ensemble act?
To find the duration of each ensemble act, we can substitute the values of x = 6 and y into either of the original equations. Let's use the first equation for simplicity:
14x + 2y = 110
14(6) + 2y = 110
84 + 2y = 110
2y = 110 - 84
2y = 26
y = 26/2
y = 13
Therefore, each ensemble act lasts for 13 minutes.