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Systems of linear equations
Advanced Level Pure Mathematics
Advanced Level Pure Mathematics
9
Algebra
Chapter 9
Systems of Linear Equations
9.1
Introduction and Existence and Uniqueness of Solution
2
9.3
Gaussian Elimination
7
9.4
Solutions of Systems of Linear Equations
8
9.5
Solution of a Homogeneous System of Equations
13
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Systems of linear equations
Advanced Level Pure Mathematics
9.1
Introduction and Existence and Uniqueness of Solution
Consider three equations in three unknowns, i.e.
a11 x1

a 21 x1
a x
 31 1
i.e
 a12 x 2
 a 22 x2
 a32 x 2
 a13 x3
 a 23 x3
 a33 x3
 b1
 b2
 b3
In Matrix From.
 a11

 a 21
a
 31
a12
a 22
a32
a13  x1   b1 
   
a 23  x 2    b2 
a33  x3   b3 
The system of three linear equations may be rewritten as
AX  B
If B  0 , the system is called a non-homogeneous system of linear equations.
For A  0 , A 1 exist.
The system becomes AX  B
X  A1 B

Three linear equations will have a unique solution. In this case, three linear equations are said to be
linearly independent.
Theorem 9-1 Let A be a square matrix. If A is a non-singular matrix, i.e. det A  0 , then the system of
linear equations AX  B has a unique solution given by X  A1 B .
Example 1
Use the method of inverse matrix to solve the system of equations
3 x  2 y  4

 x y  3
Example 2
Solve the system of equations
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2 x  y  z  4

4 x  7 y  z   11
4 x  y  7 z   3

Example 3
Example 4
2 x  y  z  4  0

Solve 4 x  7 y  z  11  0
4 x  y  7 z  3  0

Discuss the number of solutions to the following systems of linear equations:
 x  2y  3
 x  2y  3
(a)
(b)


2 x  4 y  4
2 x  4 y  6
Theorem 9.2 Let A be a n  n matrix. If det A  0 , then the linear system Ax  b has no solution
or infinitely many solutions.
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Systems of linear equations
Advanced Level Pure Mathematics
Example 5
Consider the following system of linear equations:
(a  1) x  y  z  1

 x  (a  1) y  z  a
 x  y  (a  1) z  a 2

Determine the condition that the system has a unique solution.
Theorem 9-3 Cramer's Rule
If A  0 , then
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Systems of linear equations
Advanced Level Pure Mathematics
x1 
b1
a12
a13
a11
b1
a13
a11
a12
b1
b2
b3
a 22
a32
a 23
a33
a 21 b2
a31 b3
a 23
a33
a 21
a31
a 22
a32
b2
b3
A
x2 
A
x3 
A
Example 6
4 x  y  2 z  15

Solve  x  2 y  3z  9
2 x  y  z  0

Example 7
5 x  2 y  13
Solve 
2 x  3 y  9
Example 8
2 x  5 y  4 z  2

Show that the system  x  9 y  7 z  5 has a unique solution if k  8 .
 kx  3 y  2 z  2k

Hence, solve the system by the Cramer's rule.
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Systems of linear equations
Advanced Level Pure Mathematics
9.3
Gaussian Elimination
Elementary Transformation
The elementary operations ( or elementary transformations ) in the process of elimination are:
(1)
Interchange of two equations;
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Systems of linear equations
Advanced Level Pure Mathematics
(2)
Multiplication of an equation by a non-zero scalar;
(3)
Addition of a scalar multiple of any equation to another equation.
If the system of equations is written in matrix form, we have the following corresponding elementary row
operations for the matrices:
(1)
Interchange of any rows;
(2)
Multiplication of any row by a non-zero scalar;
(3)
Addition of a scalar multiple of any row to another row.
Example 9
By using Gaussian elimination, solve the system of linear equations
4 x  y  2 z  15

 x  2 y  3z  9
2 x  y  z  0

Example 10 By using Gaussian elimination, solve the system of linear equations
4y
z  5


2 x  y  3z  13
 x  2 y  2z  3

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Systems of linear equations
Advanced Level Pure Mathematics
9.4
Solution of systems of linear Equations
Definition
Example
An inconsistent system of equation is one for which the solution set is the empty set
x  0
x  2 y  1

and  y  0

x  2 y  5
x  6

Definition
A consistent system of equations is one for which there exists a non-empty solution set.
Example
2
5 x  4 y 

 x  2 y   1.7
Example
 x  2y  1
has infinite many solutions

2 x  4 y  2
Definition
If the solution set of a consistent system of equations contains one and only one element, the
system is said to have unique solution.
Definition
If the solution set of a consistent system of equations contains more than one element, the
system is said to have non-unique solution set ( infinitely many solution )
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Systems of linear equations
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4 x  6 y  z  2

Example 11 Solve 2 x  y  4 z  3
3x  2 y  5 z  8

Determine whether the system is consistent.
If the system is consistent, solve it.
 1  1  3
 x
9


 
 
Example 12 Let A    1  3 11  , X   y  and B   7  .
 1  13 21 
 53 
z


 
 
(a)
Find A . What conclusion can you make for the system AX  B .
(b)
Solve it.
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Systems of linear equations
Advanced Level Pure Mathematics
x y z  5

Example 13 Solve 4 x  3 y  z  2
5 x  3 y  z   11

 2 y  4z  1
 x

Example 14 Solve  2 x  4 y  8 z  2
 3x  6 y  12 z   3

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Systems of linear equations
Advanced Level Pure Mathematics
 x  2 y  3z
2 x  3 y  2 z

Example 15 Solve 
4 x  y  z
4 x  y  4 z




2
5
2
1
 x  y  3z  k

Example 16 Given the system 6 x  7 y  5 z  2 has infinitely many solutions .
4 x  5 y  hz  1

Find h, k and the solution of the system.
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Systems of linear equations
Advanced Level Pure Mathematics
 px  y  z  6

3x  y  11z  6
2 x  y  4 z  q

Example 17
(a)
(b)
9.5
Find the condition of
(i)
a unique solution
(ii)
no solution
(iii)
infinitely many solutions
x y z
3x  y  11z

Hence solve 
2 x  y  4 z

xz
6




6
6
9
0
Solution of a Homogeneous System of Equations
From Theorem 9.1 and Theorem 9.2, the solution of non-homogeneous system of n linear equations in n
unknowns.
AX  B
has 3 possibilities:
(1)
If det A  0 , the system has a unique solution.
(2)
If det A  0 , has no solution or infinitely many solutions.
On the other hand, the homogenous system
AX  0
always has zero solution( trivial solution). Hence, there are only two possible cases for the solution of a
system of homogeneous equations:
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Systems of linear equations
Advanced Level Pure Mathematics
(1)
(2)
If det A  0 , the system has only zero solution (trivial solution).
If det A  0 , the system has non-zero solutions (non-trivial solution), i.e. it has infinitely many
solutions.
For AX  0
a 11x 1

a 21x 1
a x
 31 1
 a 12 x 2
 a 22 x 2
 a 32 x 2
 a 13 x 3
 a 23 x 3
 a 33 x 3
 0
 0
 0
x 1  x 2  x 3  0 is the solution of system.

(1)
x 1  x 2  x 3  0 is Trivial Solution
det A  0
A 1 exist

AX  0
X  A 1 0
X 0

(2)
the system has trivial solution.
det A  0
Example 18

the system has non-trivial solution.
x y z  0

3x  y  2 z  0
2 x  2 y  z  0

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Systems of linear equations
Advanced Level Pure Mathematics
x y z  0
Example 19 Find the non-trivial solution for 
3x  2 y  z  0
 x  ky  3z  0

Example 20 If 3x  2 y  2 z  0 ,
2 x  3 y  kz  0

find the values of k such that the system has
(a)
trivial solution
(b)
non-trivial solution and the solution set.
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Systems of linear equations
Advanced Level Pure Mathematics
Example 21 Consider the system of linear equations
 2z
 0
2 x  y

(*) x
 (k  1) z  0
 kx  y
 4z
 0

Suppose (*) has infinitely many solutions.
(a)
Find k .
(b)
Solve (*).
Example 22 Consider the following system of linear equations:
3x  y  z  1

(*)2 x  4 y  5 z  1 ,
4 x  2 y  7 z  c

where c  R .
Suppose (*) is consistent. Find c and solve (*).
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Systems of linear equations
Advanced Level Pure Mathematics
2001#9
Example 23 Consider the system of linear equations
 x  y  z  k

( S ) : x  y  z  1 where  , k  R .
3x  y  2 z   1

(a)
Show tat (S ) has a unique solution if and only if   0 and   2 .
(b)
For each of the following cases, determine the value(s) of k for which (S ) is
consistent. Solve (S ) in each case.
(c)
(i)
  0 and   2 ,
(ii)
  0,
(iii)
  2.
z  0
x

If some solution ( x, y, z ) of 
y z  1
3x  y  2 z   1

satisfies ( x  p) 2  y 2  z 2  1 , find the range of values of p .
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