Olin College of Engineering
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10-1-2010
Fall 2010 MTH 2120: Linear Algebra: Course
Materials: Lesson 1-6
Andrea Rubiano
Franklin W. Olin College of Engineering, [email protected]
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Rubiano, Andrea, "Fall 2010 MTH 2120: Linear Algebra: Course Materials: Lesson 1-6" (2010). All Course Material - Olin Course
Repository. Paper 27.
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MTH2120 Linear Algebra
1.6
Fall2010
Applications of linear systems
Practice: 7, 12
Required: 14
1.6.1
Balancing chemical equations
The number of atoms on the left should match the corresponding number of atoms on the
right, for example
2N a3 P O4 + 3Ba(N O3 )2 → Ba3 (P )4 )2 + 6N aN O3 .
In this case, the number of atoms on both sides are:
N a= 6
P = 2
O =26
Ba= 3
N = 6.
EXAMPLE 1 : Balance the following chemical equation, in which boron sulfide reacts
with water to form boric acid and hydrogen sulfide gas. The unbalanced equation is
B2 S3 + H2 O → H3 BO3 + H2 S.
Solution: We need to find x1 , · · · x4 such that we balance the equation
(x1 )B2 S3 + (x2 )H2 O → (x3 )H3 BO3 + (x4 )H2 S.
Set up a vector equation that describes the number of atoms of each type in the reaction.
There are 4 different types of atoms (B, S, H, O):
 
 
2
0
 
 
 3 
 0 

 
B2 S3 : 
 0  , H2 O :  2  , H3 BO3
 
 
0
1
To balance the equation we must solve:
 

2
0
 

 3 

 + x2  0
x1 
 0 
 2
 

0
1



1


 
 0 

:
 3  , H2 S
 
3
1


0
0

 
 1 

:
 2 .
 
0


 
 

 0 
 
 = x3   + x4  1  ,

 3 
 2 

 
 
3
0
1
MTH2120 Linear Algebra
Fall2010
or equivalently, moving all terms to the left

 
 
0
2

 
 

 0 
 3 


 
x1 
 0  + x2  2  − x3 
 

 
1
0
1


0


0

 

 1 

0 
 − x4   =  0



 0
3 
 2 

3
0
0



.


The augmented matrix is
Row reduction of the augmented matrix leads

1 0 0

 0 1 0

 0 0 1

0 0 0
to the reduced echelon form

− 31 0

−2 0 
.
− 32 0 

0 0
The general solution of this linear system is
x1 =
, x2 =
, x3 =
, x4 =
.
But, since the variables represent the number of atoms, we have to choose a nonnegative,
integer answer. This leads (usually the smallest possible values are taken) to
x1 =
1.6.2
, x2 =
, x3 =
, x4 = 3.
Network Flow
Flow of some quantity through a network. A network consist of a set of points called nodes,
with arcs called branches connecting some of the nodes.
The basic assumptions of a network flow are (i.) the total flow into the network equals the
total flow out of the network, and (ii. ) the total flow into a node equals the total flow out
2
MTH2120 Linear Algebra
Fall2010
of the node.
EXAMPLE 2 : Find the general flow pattern of the network shown below. If all the flows
are nonnegative, what is the largest value for x3 ?
Solution:
Flow at A:
x1 + x3 = 20
Flow at B:
x2 = x3 + x4
Flow at C:
80 = x1 + x2
Total flow:
80 = 20 + x4 .
The corresponding system of linear equations is:

1 0
1 0 20



 0 1 −1 0 60 


;
0 1 60 
 0 0

0 0
0 0 0
Row reduction of the augmented matrix leads lo the solution
x1 =
, x2 =
, x3 is free , x4 = 60.
Now, since all the flows need to be nonnegative the largest value for x3 will be
.
3
MTH2120 Linear Algebra
Fall2010
EXAMPLE 3 : Consider the chemical reaction
CO2 + H2 ⇒ CH4 + H2 O.
Explain why it is always possible to balance this equation. Describe geometrically the
solution set. Find a possible solution to balance this equation.
EXAMPLE 4 : Find the general flow pattern of the traffic network shown below. If all
the flows are nonnegative, find the minimum an largest values for x1 , x2 , x3 and x4 .
4