Solve the system using elimination.
19x - 9y = 2
x - 9y = -34
(Type an ordered pair.)
To solve the system using elimination, we can multiply the second equation by 19 and then subtract it from the first equation.
(19x - 9y) - 19(x - 9y) = 2 - 19(-34)
19x - 9y - 19x + 171y = 2 + 646
162y = 648
y = 648/162
y = 4
Now we can substitute this value of y back into either equation to solve for x. Let's use the second equation:
x - 9(4) = -34
x - 36 = -34
x = -34 + 36
x = 2
Therefore, the solution to the system is (x, y) = (2, 4).
Your school's talent show will feature 14 solo acts and 3 ensemble acts. The show will last 134 minutes. The 7 solo performers judged best will give a repeat performance at a second 85 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's assign variables to the unknown quantities:
Let x represent the number of minutes each solo act lasts.
Let y represent the number of minutes each ensemble act lasts.
From the given information, we can form two equations:
1) 14x + 3y = 134
There are 14 solo acts that last x minutes each, and 3 ensemble acts that last y minutes each. The total duration of these acts is 134 minutes.
2) 7x + 3y = 85
Only the 7 solo performers judged best will give a repeat performance at the second show, which lasts 85 minutes. The duration of the ensemble acts is still y minutes each.
Now we can solve the system of equations.
To eliminate y, we can multiply the first equation by 3 and the second equation by -3:
3(14x + 3y) = 3(134) -> 42x + 9y = 402
-3(7x + 3y) = -3(85) -> -21x - 9y = -255
Adding these two equations eliminates y:
42x + 9y + (-21x - 9y) = 402 + (-255)
21x = 147
x = 147/21
x = 7
Now we can substitute this value of x back into one of the equations to solve for y. Let's use the first equation:
14(7) + 3y = 134
98 + 3y = 134
3y = 134 - 98
3y = 36
y = 36/3
y = 12
Therefore, each solo act lasts 7 minutes, and each ensemble act lasts 12 minutes.
(a) Each solo act lasts 7 minutes, and each ensemble act lasts 12 minutes.
(b) The second show will feature the 7 solo performers judged best (7 acts) giving a repeat performance, which lasts 85 minutes, and the 3 ensemble acts, which also last y minutes. Since each solo act lasts 7 minutes, the duration of the solo acts in the second show is 7 (acts) * 7 (minutes/act) = 49 minutes.
The total duration of the second show is 85 minutes, and the duration of the ensemble acts is y minutes. Therefore, the equation is:
49 + 3y = 85
To solve for y:
3y = 85 - 49
3y = 36
y = 36/3
y = 12
Therefore, the second show will also feature the 3 ensemble acts, each lasting 12 minutes.
b) Solve the system from part (a).
(Type an ordered pair.)
From part (a), we determined that each solo act lasts 7 minutes and each ensemble act lasts 12 minutes.
Therefore, the ordered pair that represents the solution to the system is (7, 12).
part c) How long is each solo act?
[ ] minute(s)
From part (a), we determined that each solo act lasts 7 minutes. Therefore, each solo act is 7 minutes long.
part (d) How long is each ensemble act?
[ ] minute(s)