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 x2
 a13 x3
 b1
 a 22 x2
 a32 x2
 a 23 x3
 a33 x3
 b2
 b3
In Matrix From.
 a11

 a 21
a
 31
a12
a 22
a32
a13  x1   b1 
   
a 23  x2    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
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Systems of linear equations
Advanced Level Pure Mathematics
Example 2
Solve the system of equations
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
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Systems of linear equations
Advanced Level Pure Mathematics
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.
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.
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Systems of linear equations
Advanced Level Pure Mathematics
Theorem 9-3 Cramer's Rule
If A  0 , then
x1 
b1
b2
b3
a12
a 22
a32
a13
a 23
a33
A
x2 
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
a11 b1
a 21 b2
a31 b3
A
a13
a 23
a33
x3 
a11
a 21
a31
a12
a 22
a32
b1
b2
b3
A
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Systems of linear equations
Advanced Level Pure Mathematics
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;
(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)
(2)
(3)
Interchange of any rows;
Multiplication of any row by a non-zero scalar;
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

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Systems of linear equations
Advanced Level Pure Mathematics
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

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
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Systems of linear equations
Advanced Level Pure Mathematics
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 )
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.
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Systems of linear equations
Advanced Level Pure Mathematics
 1  1  3
9
 x


 
 
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.
x y z  5

Example 13 Solve 4 x  3 y  z  2
5 x  3 y  z   11

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Systems of linear equations
Advanced Level Pure Mathematics
 2 y  4z  1
 x

Example 14 Solve  2 x  4 y  8 z  2
 3x  6 y  12 z   3

 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
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Systems of linear equations
Advanced Level Pure Mathematics
 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.
 px  y  z  6

 3x  y  11z  6
2 x  y  4 z  q

Example 17
(a)
(b)
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
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Systems of linear equations
Advanced Level Pure Mathematics
9.5
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:
(1)
If det A  0 , the system has only zero solution (trivial solution).
(2)
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 13 x 3
 0
 a 22 x 2
 a 32 x 2
 a 23 x 3
 a 33 x 3
 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

the system has non-trivial solution.
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Systems of linear equations
Advanced Level Pure Mathematics
Example 18
x y z  0

3x  y  2 z  0
2 x  2 y  z  0

x y z  0
Example 19 Find the non-trivial solution for 
3x  2 y  z  0
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Systems of linear equations
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 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.
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 (*).
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Systems of linear equations
Advanced Level Pure Mathematics
Example 22 Consider the following system of linear equations:
3 x  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
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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