ES Math project
Done by:
Abdulrhman Ahmed
Mohammed Ahmed
Hamad Jumaa
Mohammed Abdullah
task1
• The problem has been solved by Gaussian
Elimination. Which means at the end we are
going to get
• First right the equation as augmented matrix
After that multiply ½ by the first row
• multiply the first raw by -1 and add the second
row. (-1R1+R2)
multiply the first raw by 3 and subtract it from the third raw. (3R1-R3)
• Multiply the second row by 2/5. (2/5 R2)
Multiply the second row by ½ and add the third row. ( ½ R2+R3)
• Multiply the third row by 5/18. (5/18 R3)
We followed Gaussian Elimination and we got
• Z= 1500
• y+1/5(1500)=1300
y+300= 1300
y= 1000
• X + 1/2 (1000) + 3/2 (1500)=4000
x + 500 + 2250 = 4000
x + 2750 = 4000
x= 1250
Task 2
You own two stores that sell household
appliances. The matrices below show revenue
and expenses for three months at each store.
REVENUE ($)
EXPENSES ($)
a. Write a matrix that shows the
monthly profit for each store.
b. Which store had higher overall profits during the
three- month period?
c. Which store lost money? In which
month?
Task 3
A. Members of the Cooking Club entered the Culinary Challenge. In
this contest, the score for each entry is multiplied by an assigned
degree of difficulty.
Cooking Club Members Score
Culinary Challenge Degrees Of Difficulty
Appetizer
Main
Course
Dessert
Beth
25
38
28
Jon
35
29
37
Lupe
20
31
39
Amy
40
32
36
Beth
Jon
Lupe
Amy
Appetizer
3.1
2.0
3.5
1.5
Main Course
2.1
1.8
3.7
2.8
Dessert
2.3
2.4
3.0
3.5
1. Display each table as a matrix. Matrix S should show
the scores and matrix D should show the degree of
difficulty
S=
25
38
28
35
29
37
20
31
39
40
32
36
D=
3.1
2.0
3.5
1.5
2.1
1.8
3.7
2.8
2.3
2.4
3.0
3.5
2) Write an equation using S, D and product matrix P you
could use to evaluate the final scores.
S‘ D =P
25
38
28
35
29
37
20
31
39
40
32
36
x
3.1
2.0
3.5
1.5
2.1
1.8
3.7
2.8
2.3
2.4
3.0
3.5
=P
S‘ D =P
4x3
3x4
4x4
Because the number of columns in “S“ matrix is the same
number of the rows in “D“ matrix
4) Write the product matrix P
25
38
28
35
29
37
20
31
39
40
32
36
p=
x
3.1
2.0
3.5
1.5
2.1
1.8
3.7
2.8
2.3
2.4
3.0
3.5
221.7
185.6
312.1
241.9
254.5
211
340.8
263.2
216.8
189.4
301.7
253.3
274
224
366.4
275.6
=p
5) Roger is writing a story for the school newspaper
about the Culinary Challenge. Explain how he can
use P to find the final scores for his story
Roger will add the number of one row and the total will
be the final scores of the person who Roger picked.
6) List the contestants and their final scores, in
descending order.
1)
Lupe 1140
2)
Jon 1069.5
3)
Beth 961.3
4)
Amy 961.2
B] Use the information provided to decode the three-word
message made up of three-letter words encrypted in the
matrices below.
161
170
113
145
150
109
62
61
45
The three-letter words are multiplied by the matrix
The message can be decoded by multiplying the
coded message by the inverse of the encoding
matrix.
M=
6
5
2
5
5
2
2
2
1
1. What is the inverse matrix?
Det(M)
6
5
2
5
5
2
2
2
1
6(5-4) -5(5-4) +2(10-10) = 1
Det ( M ) = 1
+
-
+
5
2
2
1
5
2
2
1
5
2
5
2
-
+
-
5
2
2
1
6
2
2
1
6
2
5
2
+
-
+
5
5
2
2
6
5
2
2
6
5
5
5
M-1 =
1
-1
0
-1
2
-2
0
-2
5
161
145
2. What word is coded by
1
-1
0
-1
2
-2
0
-2
5
161
x
145
62
=
62
16
P
5
E
20
T
?
170
3. What word is coded by
150
61
1
-1
0
-1
2
-2
0
-2
5
170
x
150
61
=
20
T
8
H
5
E
?
113
4. What word is coded by
?
109
45
1
-1
0
-1
2
-2
0
-2
5
113
x
109
45
=
4
D
15
O
7
G
5. Write the message :
PET
THE
DOG
Task 4
Row Operations and Augmented Matrices
At the annual craft show, the Ceramics Club
members sell mugs for $6.00, bowls for $5.50,
and plates for $9.50. They have for sale one
more bowl than the number of plates and 3
times as many mugs as plates. They sold
everything for a total of $236.50. How many of
each item did they sell?
1. Write a system of equations to represent the
Problem , using m, b, and p for the variables.
1)
m$6+b$5.50+p$9.50=$236.50
1+p=b
b-p=1
3p=m
3p-m=0
2. Write the augmented matrix for the system
of equations.
3. Find the reduced row-echelon form of the
augmented matrix.
6
0
1
5.50
1
0
9.50
-1
-3
236.50
1
0
=
1
0
1
11/12
1
0
19/12
-1
-3
473/12
1
0
=
1
0
0
11/12
1
-11/12
19/12
-1
-55/12
473/12
1
-473/12
R1x(1/6)
(R1x-1)+R3
=
(R2x-11/12)
+R3
=
R3x(-2/11)
1
0
0
11/12
1
0
19/12
-1
-11/2
473/12
1
-473/12
1
0
0
11/12
1
0
19/12
-1
1
473/12
1
7
4. How many of each item did the Ceramics Club
sell?
3)
p=7
b-p=1
6m+5.5b+9.5p=236.5
b-7=1
b=1+7
b=8
6m+44+66.5=236.5
6m=126
m=21
Ceramics Club sell 1 plate, 8 bowls, and 21 mugs
Task 5
Determinants and Cramer’s Rule
As Kristin prepares for a triathlon, she makes a chart of her exercise time, along
with the calories burned each day. Part of her chart is shown in the table below.
How many calories per hour does she burn for each activity?
Triathlon Training Record
Day
Swimming(h)
Cycling(h)
Running(h)
Calories
Burned
Friday
1.5
2.0
0.5
2450
Saturday
2.5
3.0
1.5
4310
Sunday
2.0
1.2
1.6
3150
1) Write a system of equations that relates Kristin’s exercises time to the
number of calories burned each day. Use s, c , and r for the calories burned
per hour for the three activities.
2) Write the coefficient matrix for the
system of equations
3) What is the value ,D , for the
determinant of the coefficient matrix?
4) Use Cramer’s rule to solve this
system of equations. Give the values
for s, c, and r
• First we find the determinant of the matrix
4) Use Cramer’s rule to solve this
system of equations. Give the values
for s, c, and r
• Find |A1|
4) Use Cramer’s rule to solve this
system of equations. Give the values
for s, c, and r
• Find |A2|
4) Use Cramer’s rule to solve this
system of equations. Give the values
for s, c, and r
• Find |A3|
4) Use Cramer’s rule to solve this
system of equations. Give the values
for s, c, and r
The Solution
is
( 869 , 434 , 557 )
Task 6
• 1) Write a system of equations, using p, t, and
f as the cost per ounce of each kind of salad.
2) Set up the matrix equation, Ax=B
• 3) Find the determinant of matrix A
• 4) Find A-1
• 1/3 R1
• -1 R1 + R2
• -2 R1 + R3
• 3/4 R3
• -8/3 R3+R2
• -1/3 R3+R1
• -1R2+R1
• 5) Solve X = A-1 B for X
• 6) What is the price per ounce for each kind of
salad?
Task7
1- Search on the internet for an economic
applications that need to be solved by the
matrices.
Ram, Shyam and Mohan purchased biscuits of different brands P, Q and R.
Ram purchased 10 packets of P, 7 packets of Q and 3 packets of R.
Shyam purchased 4 packets of P, 8 packets of Q and 10 packets of R.
Mohan purchased 4 packets of P, 7 packets of Q and 8 packets of R.
If brand P costs Rs 4, Q costs Rs 5 and R costs Rs 6 each,
then using matrix operation to find the amount of money spent by these persons
individually.
We gather the information then we make this tables which we will
change them to matrices and solve them.
P
Q
R
Ram
10
7
3
Shyam
4
8
10
Mohan
4
7
8
P
4
Q
5
R
6