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Question One
It is 2011 and the Minnesota State Fair is just around the corner. Ryan can’t wait to go and stand in line for all of his favorite state fair foods. He always make sure that he stops at the following booths: Deep Fried Candy Bar $3.50, Cotton Candy $3.25, Corn on the Cob $4.00, Pronto Pup $3.75, and the Pork Chop on a Stick $5.25. Ryan heard from Lena that all the food booths at the state fair are increasing their prices by 7%. If this is the case how much will he now have to pay for all of his favorite foods?
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Helpful Advice
For this problem using scalar multiplication
would be a very useful tool.
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Step 1: Make a Matrix
2010Prices
Deep Fried Candy Bar 3.50Cotton Candy 3.25Corn on the Cob 4.00Pronto Pups 3.75Pork Chop on a Stick 5.25
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Scalar * Matrix 1 + Matrix 1 Setting up the matrix problem
Scalar *
a
=
Scalar * ab Scalar * bc Scalar * cd Scalar * de Scalar * e
Scalar Multiplication
Review
Scalar Addition Review
a
+
f
=
a + fb g b + gc h c + hd i d + ie j e + j
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Run the numbers
0.08 *
3.50
+
3.503.25 3.254.00 4.003.75 3.755.25 5.25
0.28
+
3.50
=
3.780.26 3.25 3.510.32 4.00 4.320.30 3.75 4.050.42 5.25 5.67
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Another Route 1.08 *
3.503.254.003.755.25
By having your scalar equal to 1.08 you are eliminating the need to add the percentage increase to the original amount – this new scalar will do just that!
Just multiple the matrix by the scalar and you will get your result immediately!
=
3.783.514.324.055.67
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Final Solution
2010Prices
2011 Price
sDeep Fried Candy Bar 3.50 3.78Cotton Candy 3.25 3.51Corn on the Cob 4.00 4.32Pronto Pups 3.75 4.05Pork Chop on a Stick 5.25 5.67
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Question Two
Solve the following system of linear equations.
X + 4Y = 32X + 9Y = 5
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NotationRow switching : A row within the matrix can
be switched with another row. R1 R2 Row multiplication: Each element in a row can be multiplied by a non-zero constant.k* R1 R1, k ≠ 0Row addition: A row can be replaced by the sum of that row and a multiple of another row. R1 + kR2 R1, k ≠ 0
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Goal
1 0 0 #1
0 1 0 #2
The ultimate goal is to get our matrix to look like the one to the left and then the values in the #1, #2 locations will be the solutions to the problem for x and y
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Using row operations here we go….Step 1: Set up the augmented matrix
1 4 32 9 5
Step 2: First Operation
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Step 2: First Operation
-2R1 + R2 R2
1 4 30 1 -1
Notice no change was made to R1
(row 1)
R1 = Row One and R2 = Row Two
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Step 3: Second Operation
-4R2 + R1 R1
1 0 70 1 -1 Notice no
change was made to R2
(row 2)
R1 = Row One and R2 = Row Two
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Step 4: Finalize Solution
1 0 70 1 -1
X Y
X = 7 Y = -1
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Step 5: Double check the solution you found
X + 4Y = 32X + 9Y = 5
Original Equations
(7) + 4(-1) = 32(7) + 9(-1) = 5
It Works!
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Question Three
Solve the following system of linear equations.
X + Y - Z = -22X – Y + Z= 5-X+2Y+2Z = 1
Same idea but now we have to work with the three variables.
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Goal
The ultimate goal is to get our matrix to look like the one to the left and then the values in the #1, #2, #3 locations will be the solutions to the problem for x and y and z
1 0 0 #10 1 0 #20 0 1 #3
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Using row operations here we go….Step 1: Set up the augmented matrix
1 1 -1 -22 -1 1 5-1 2 2 1
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Step 2: First Operation
-2R1 + R2 R2
Notice no change was made to R1
and R3
R1 = Row One and R2 = Row Two and R3 = Row Three
1 1 -1 -20 -3 3 9-1 2 2 1
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Step 3: Operation
R1 + R3 R3
Notice no change was made to R1
and R2
R1 = Row One and R2 = Row Two and R3 = Row Three
1 1 -1 -20 -3 3 90 3 1 -1
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Step 4: Operation
R2 ÷ -3 R2
Notice no change was made to R1
and R3
R1 = Row One and R2 = Row Two and R3 = Row Three
1 1 -1 -20 1 -1 -30 3 1 -1
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Step 5: Operation
-3R2 + R3 R3
Notice no change was made to R1
and R2
R1 = Row One and R2 = Row Two and R3 = Row Three
1 1 -1 -20 1 -1 -30 0 4 8
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Step 6: Operation
R3 ÷ 4 R3
Notice no change was made to R1
and R2
R1 = Row One and R2 = Row Two and R3 = Row Three
1 1 -1 -20 1 -1 -30 0 1 2
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Step 7: Finalize Solution
1 1 -1 -20 1 -1 -30 0 1 2
X + Y - Z = -2Y - Z= -3
Z = 2
Now:
Then: Y - Z = -3Y - 2= -3
Y = -1
Lastly: X + Y - Z = -2X + (-1) – 2 = -2
X-3 = -2X= 1
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Step 8: Double check the solution you found
X + Y - Z = -22X – Y + Z= 5-X+2Y+2Z = 1
Original Equations
(1) + (-1) – (2) = -22(1) – (-1) + (2)= 5-(1)+2(-1)+2(2) = 1
It Works!!
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Question Four
Now to wrap it all up we want to determine how much it actually costs to get into the fair and enjoy this fun get together! As you saw on the clip, one day at the fair just isn’t enough!! At the gates on opening day they are selling one day passes as well as bonus three day passes. We were only able to collect the data for 2 particular gates on this day. Gate A sells 2,090 day passes and 980 three day passes. Gate B sells 1,807 day passes and 712 three day passes. If Gate A collected $31,290 and Gate B collected $24,753, how much does it cost to by a one day pass and a three day pass?
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Helpful Advice
For this problem finding the determinate and
the inverse would be a very useful tool.
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Set up a Matrix Matrix 1: A B
C DMatrix Format
1-day 3-day Total $Gate A 2090 980 $31,290Gate B 1807 712 $24,753
Matrix specific to this example
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Finding the determinantDeterminant = A*D - B*C
2090 9801807 712
A BC D
2090 * 712 - 980 * 1807
Det A = -282,780
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Finding the inverse of a matrix Inverse Matrix = 1 * D -B
Det A -C A
-3.5363E-06 * 712 -980 = -0.0025 0.0035
-1807 2090 0.0064 -0.0074
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Multiplying inverse and solution to find unknown
A B x E
C D F
= (A * E) + (B * F)
(C * E) + (D * F)
-0.002518 0.003466 X 31290
0.006390 -0.007391 24753
= $7
$17
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Double check the solutions to make sure they do indeed work in the problem
2090 980 X $7
1807 712 $17
= $31,290$24,753
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Final Solution
One Day Pass $7.00Three Day Pass $17.00
In the future you can go to this Excel document to plug in your matrix information and have it automatically calculate