Download Problems needing assistance

Problems needing assistance
1. What is the solution of the system? Type an ordered pair
5x+3y= -11
7x-2y= 17
(1)
(2)
We solve the system by elimination method to eliminate y.
Multiply 2 to each side of equation (1)
10x + 6y = -22
(3)
Multiply 3 to both sides of the equation (2)
21x – 6y = 51
(4)
(3) + (4)
31x = 29
29
x
31
Substituting 29/31 to equation (1) to solve for y:
29
5*
 3 y  11
31
145
486
3 y  11 

31
31
162
y
31
 29 162 
Therefore, the solution of the system is:  , 
.
 31 31 
2. Hockey team receives 2 points when they win and 1 point when they tie. One season, a
team won a championship with 60 points. They won 12 more games than they tied. How
many wins and how many ties did the team have?
Let x be the number of games they won and y be the number of the games they tied.
They won 12 more games than they tied translates to:
x = y + 12
For x games they won, they received 2x points; for y gamed they tied, they received 1 *y
points. The total points are 60. so the equation is
2x + y = 60
Therefore, we have a linear system of:
x = y + 12
(1)
2x + y = 60
(2)
Substituting (1) to (2) gives
2(y + 12) + y = 60
2y + 24 + y = 60
3y = 36
y = 12
then x = y + 12 = 12 + 12 = 24
so the team won 24 games and tied 12 games.
3. Solve by the elimination method. What is the solution of the system? Type an ordered
pair.
8x-9y= 32.5
7y-2x= -10.5
(1)
(2)
By using the elimination method, we need to have the coefficients of one of the two
variables in the two equations equal or in opposite signs.
We can eliminate y first.
Multiply 7 to equation (1)
56x – 63y = 227.5
(3)
Multiply 9 to equation (2)
63y – 18x = -94.5
(4)
Add (3) and (4) together:
56x – 63y + 63y – 18x = 227.5 – 94.5
38x = 133
x = 3.5
substituting 3.5 for x to equation (2) to solve for y:
7y – 2 * 3.5 = -10.5
7y = -3.5
y = -0.5
so the solution is  3.5, 0.5
4. What is the solution of the system? Type an ordered pair.
0.05x+ 0.25y= 66
(1)
0.15x+0.05y= 72
(2)
Using the elimination method to eliminate x:
Multiply 3 to both sides of (1):
0.15x + 0.75y = 198
(3)
(3) – (2)
0.15x + 0.75y – (0.15x + 0.05y) = 198 – 72
0.7y = 126
y = 180
substituting 180 for y to equation (1) to solve for x:
0.05x + 0.25 * 180 = 66
0.05x = 66 – 45
0.05x = 21
x = 420
solution: (420, 180)
5. The perimeter of a rectangle is 240 inches. The length exceeds the width by 44 inches.
Find the length and the width.
Length: L; width: W, both units: inch
The perimeter of the rectangle is p = 2* L + 2 *W
2L + 2W = 240
The length is 44 inches longer than the width, so
L = W + 44
The linear system is
2L + 2W = 240
(1)
L = W + 44
(2)
Using the substitution method, substituting (2) to (1)
2(W + 44) + 2W = 240
2W + 88 + 2W = 240
4W = 152
W = 38
Then L = W + 44 = 38 + 44 = 82
The length is 82 inches and the width is 38 inches.
6. A disc jockey must play 14 commercial spots during 1 hour of a radio show. Each
commercial is either 30 seconds or 60 seconds long. If the total commercial time during 1
hour is 11 minutes, how many 30 second commercials were played that hour? How many
60 second commercials?
Let x and y be the number of 30 s and 60 s commercials played during the 1 hour
program, respectively.
The first equation is simple:
The total number of commercials is 14, so
x + y = 14
The time for x 30 s (0.5 minutes) commercial is 0.5x minutes.
The time used for y 60 s (1 minute) commercial is 1 * y = y minutes.
The total commercial time is 11 minutes. So
0.5x + y = 11
The linear system is
x + y = 14
(1)
0.5x + y = 11
(2)
(1) – (2)
0.5x = 3
x=6
then substituting 6 for x to solve for y in equation (1):
6 + y = 14
y=8
Therefore, the DJ played 6 30-second commercials and 8 60-second commercials