HW7.4 Matrix Application
Problems:
Select and state 3 suitable variables. Convert the word problem into a 3 equations that
relate the variables. Convert the equations into a 3x3 augmented matrix. Then, solve the matrix.
1.
Derrick invested $10,000 for one year in three different investments. The investments paid
simple annual interest of 5%, 6%, and 7%, respectively, and he received a total of $610 interests
for the year. He invested $3000 more at 5% than at 6%. Find the amount invested at each rate.
𝑥 = amount invested at 5%
𝑦 = amount invested at 6%
𝑧 = amount invested at 7%
𝑥 + 𝑦 + 𝑧 = 10000
0.05𝑥 + 0.06𝑦 + 0.07𝑧 = 610
𝑥 − 𝑦 = 3000
1
[. 05
1
1
. 06
−1
1
. 07
0
10000
610 ]
3000
(4000,1000,5000)
2.
A box of wood contains wood cut into triangular, square, and pentagonal shapes. There are 80
pieces of wood in the box, and the pieces have a total of 290 sides. If there are 10 more
triangular pieces than square pieces, find the number of pieces of wood of each shape in the
box.
𝑡 = # of triangular shapes
𝑠 = # of square shapes
𝑝 = # of pentagonal shapes
𝑡 + 𝑠 + 𝑝 = 80
3𝑡 + 4𝑠 + 5𝑝 = 290
𝑡 − 𝑠 = 10
(40,30,10)
1 1 1
[3 4 5
1 −1 0
80
290]
10
3.
A change machine contains nickels, dimes, and quarters. There are 75 coins in the machine,
and the value of the coins is $7.25. If there are 5 times as many nickels as dimes, find the
number of coins of each type in the machine.
𝑛 = # of nickels
𝑑 = # of dimes
𝑞 = # of quarters
𝑛 + 𝑑 + 𝑞 = 75
0.05𝑛 + 0.10𝑑 + 0.25𝑞 = 7.25
𝑛 − 5𝑑 = 0
1
1
1
75
[0.05 0.1 0.25 7.25]
1
−5
0
0
(50,10,15)
4.
The sum of the digits of a three digit number is 17. Four times the hundreds digit minus 5 times
the tens digit is 12. If 7 times the units digit is added to 3 times the tens digit, the result is 47.
Find the number.
𝑢 = # in units digit
𝑡 = # in tens digit
ℎ = # in hundreds digit
𝑢 + 𝑡 + ℎ = 17
−5𝑡 + 4ℎ = 12
7𝑢 + 3𝑡 = 47
548
1
[0
7
1 1 17
−5 4 12]
3 0 47
5.
The length of a rectangular shed is twice its height, and the height of the shed is one foot
greater than its width. If the base of the shed has a perimeter of 40 feet, find the dimensions of
the shed.
𝑙 = length of shed
𝑤 = width of shed
ℎ = height of shed
𝑙 − 2ℎ = 0
−𝑤 + ℎ = 1
2𝑙 + 2𝑤 = 40
1 0 −2
[0 −1 1
2 2
0
0
1]
40
(14,6,7)
6.
A bin in a grocery store contains 100 lb. of a mixture of almonds, peanuts, and raisins. Almonds
sell for $1.89 per lb., peanuts for $1.58 per lb., and raisins for $1.39 per lb. If the mixture
contains twice as many pounds of peanuts as almonds, and if the total value of the almonds
and raisins in the mixture is $93.40, how many pounds of each item does the mixture contain?
𝑎 = # of lbs of almonds
𝑝 = # of lbs of peanuts
𝑟 = # of lbs of raisins
𝑎 + 𝑝 + 𝑟 = 100
1.89𝑎 + 1.39𝑟 = 93.40
2𝑎 − 𝑝 = 0
(20,40,40)
1
1
1
[1.89 0 1.39
2
−1
0
100
93.4]
0