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)

Question

b) Solve the system from part ​(a)
[ ] ​(Type an ordered​ pair.)

Answer 1

To solve the system from part (a), we need to set up two equations based on the given information.

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

From the given information, we know that there will be 16 solo acts and 3 ensemble acts in the first show. Since the show lasts 129 minutes, we can set up the equation:

16x + 3y = 129

From the given information, we also know that in the second show, there will be 8 solo acts and 3 ensemble acts. Since the show lasts 81 minutes, we can set up the equation:

8x + 3y = 81

Now, we can solve this system of equations using any method. Let's use substitution.

From the second equation, we can rearrange it to solve for y:

3y = 81 - 8x
y = (81 - 8x) / 3

Substitute this expression for y into the first equation:

16x + 3((81 - 8x) / 3) = 129

Multiply through by 3 to get rid of the fraction:

16x + 81 - 8x = 387

Combine like terms:

8x + 81 = 387

Subtract 81 from both sides:

8x = 306

Divide both sides by 8:

x = 38.25

Now substitute this value of x into the second equation to find y:

8(38.25) + 3y = 81
306 + 3y = 81
3y = -225
y = -75

Therefore, the solution to the system is (x, y) = (38.25, -75).

Answer 2

Part C) How long is each solo act?

Answer 3

From the solution to the system of equations in part (b), we found that x = 38.25.

Therefore, each solo act lasts 38.25 minutes.