Your​ school's talent show will feature 12 solo acts and 3 ensemble acts. The show will last 111 minutes. The 6 solo performers judged best will give a repeat performance at a second 75 minute​ show, which will also feature the 3 ensemble acts. Each solo act lasts x​ minutes, and each ensemble act lasts y minutes.

Question

Write a system of equations to model the situation.

Answer 1

Let's assume that each solo act lasts x minutes and each ensemble act lasts y minutes.

First, let's determine the total duration of the solo acts in the first show. Since there are 12 solo acts, the total duration of the solo acts in the first show is 12x minutes.

Next, let's determine the total duration of the ensemble acts in the first show. Since there are 3 ensemble acts, the total duration of the ensemble acts in the first show is 3y minutes.

The total duration of the first show is given as 111 minutes. Therefore, we have the equation:

12x + 3y = 111

For the second show, the 6 solo acts judged best will give a repeat performance. Since there are 6 solo acts in the second show, the total duration of the solo acts in the second show is 6x minutes.

The total duration of the second show is given as 75 minutes. Therefore, we have the equation:

6x + 3y = 75

The system of equations to model the situation is:

12x + 3y = 111
6x + 3y = 75

Answer 2

solve that system

Answer 3

To solve the system of equations:

12x + 3y = 111
6x + 3y = 75

We can use the method of substitution or elimination.

Let's solve it using the method of elimination:

First, let's multiply the second equation by 2 to make the coefficients of y match in both equations:

2 * (6x + 3y) = 2 * 75
12x + 6y = 150

Now we can subtract the first equation from the new equation to eliminate the x variable:

(12x + 6y) - (12x + 3y) = 150 - 111
12x - 12x + 6y - 3y = 39
3y = 39

Dividing both sides of the equation by 3, we get:

y = 39 / 3
y = 13

Now, substitute the value of y into one of the original equations, let's use the first equation:

12x + 3y = 111
12x + 3(13) = 111
12x + 39 = 111

Subtract 39 from both sides:

12x = 111 - 39
12x = 72

Divide both sides by 12:

x = 72 / 12
x = 6

So the solutions to the system of equations are x = 6 and y = 13.