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Lost Price List
After a heavy rain, a fruit seller discovered that the price board in the shop had been completely
erased. Unfortunately, the backup record containing the prices of individual fruits was also lost.
However, the billing records of three customers were successfully recovered.
The recovered bills are shown below.
Customer
1
2
3
Purchase Details
2 Apples + 3 Bananas + 1 Orange = Rs. 170
3 Apples + 2 Bananas + 4 Oranges = Rs. 320
5 Apples + 1 Banana + 2 Oranges = Rs. 260
The seller wants to recover the price of each fruit using the available billing records.
Tasks
(a) Define suitable variables.
(b) Form a system of linear equations.
(c) Write the corresponding augmented matrix.
(d) Use the Gaussian Elimination Method (Reduced Row Echelon Form) to determine the
price of each fruit.
(e) Verify your answer using all three bills.
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Chemical Reaction Balancing
In chemistry, balancing a chemical equation requires ensuring that the total number of atoms
for each element is identical on both sides of the reaction.
Consider the combustion reaction of propane gas (C3 H8 ):
x1 C3 H8 + x2 O2 −→ x3 CO2 + x4 H2 O
where x1 , x2 , x3 , and x4 represent the stoichiometric coefficients.
1. Equate atomic counts: Set up the balance equations for Carbon (C), Hydrogen (H),
and Oxygen (O) to form a system of linear homogeneous equations in terms of x1 , x2 , x3 ,
and x4 .
2. Reduce system variables: Assuming 1 mole of propane reacts (x1 = 1), simplify the
linear system into a system of three equations with three unknown variables (x2 , x3 , x4 ).
3. Augmented Matrix formulation: Construct the corresponding augmented matrix [A |
B] for the three-variable system.
4. Gauss Elimination: Apply the Gauss Elimination Method (using forward elimination to reduce the augmented matrix into row-echelon form, followed by back-substitution)
to determine x2 , x3 , and x4 .
5. Balanced Equation: Write down the final balanced chemical reaction using the smallest
integer coefficients.
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3
Steady-State Temperature Distribution
An important concern in the study of heat transfer is to determine the steady-state temperature
distribution of a thin plate when the temperature around the boundary is known. Assume the
plate shown in the figure represents a cross section of a metal beam, with negligible heat flow
in the direction perpendicular to the plate.
Let T1 , . . . , T4 denote the temperatures at the four interior nodes of the mesh shown in the
figure. The temperature at a node is approximately equal to the average of the four nearest
nodes—to the left, above, to the right, and below.
For example,
T1 =
10 + 20 + T2 + T4
,
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or equivalently,
4T1 − T2 − T4 = 30.
10◦
10◦
20◦
20◦
1
2
4
3
30◦
30◦
40◦
40◦
Problem. Write a system of four linear equations whose solution gives estimates for the
temperatures T1 , T2 , T3 , and T4 .
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Transport Company
A transport company uses three types of trucks T1 , T2 , and T3 to transport three types
of vehicles V1 , V2 , and V3 . The carrying capacity of each truck in terms of the three types of
vehicles is given below:
T1
T2
T3
V1 V2 V3
1 3 2
2 2 3
3 2 2
2
Using the matrix method, find:
1. The number of trucks of each type required to transport 85, 105, and 110 vehicles of types
V1 , V2 , and V3 , respectively.
2. Find the number of vehicles of each type that can be transported if the company has 10,
20, and 30 trucks of types T1 , T2 , and T3 , respectively.
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Traffic control
A city’s traffic engineers monitor a one-way loop of four downtown intersections A, B, C, D,
connected by internal streets
A → B,
B → C,
C → D,
D → A,
whose (unknown) hourly traffic volumes are x1 , x2 , x3 , x4 , respectively. The network also exchanges traffic with the rest of the city through four external streets, whose hourly volumes
have been measured exactly, as shown below.
200
300
x1
A
B
x4
100
x2
D
x3
C
400
At each intersection, the total inflow must equal the total outflow (traffic is neither created
nor destroyed).
1. Using conservation of flow at each of the four intersections, formulate this situation as a
linear system
Ax = b
in the unknowns
x = (x1 , x2 , x3 , x4 )T .
(4)
2. Reduce the augmented matrix [A | b] to row-echelon form. Determine rank(A) and
rank([A | b]), and explain what their relationship tells you about the existence of a
solution.
(4)
3. Find the general solution of the system, expressed as a particular solution plus a scalar
multiple of a vector spanning the solution space of the associated homogeneous system
Ax = 0.
(3)
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6
Find the Secret Meeting Point at Plaksha
As part of the annual Plaksha Treasure Hunt, four students—Aarav, Diya, Kabir, and
Meera—receive four encrypted clues about the location of a secret meeting point on the Plaksha
campus.
Each student receives only one clue, and none of the clues is sufficient to determine the
meeting point independently. The organizers intentionally designed the clues so that they must
be combined to reveal the exact location.
Now decode the clues and determine the secret meeting point. Encrypted Clues:
Aarav’s Clue
“Twice the east coordinate, together with the north coordinate, the elevation, and
the checkpoint code, equals 35.”
Diya’s Clue
“The east coordinate, three times the north coordinate, twice the elevation, and the
checkpoint code together equal 48.”
Kabir’s Clue
“Three times the east coordinate, the north coordinate, twice the elevation, and
twice the checkpoint code add up to 54.”
Meera’s Clue
“The east coordinate, twice the north coordinate, the elevation, and three times the
checkpoint code together equal 42.”
Figure 1: Caption of the image
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