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).
Solve the system using elimination.
19x - 9y = 2
x - 9y = -34
(Type an ordered pair.)
9 answers
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.
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.)
(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).
Therefore, the ordered pair that represents the solution to the system is (7, 12).
part c) How long is each solo act?
[ ] minute(s)
[ ] 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)
[ ] minute(s)
From part (a), we determined that each ensemble act lasts 12 minutes. Therefore, each ensemble act is 12 minutes long.
1 answer