Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
Section 1.1: Introduction to systems of linear
equations
Dr K
Department of Mathematics and Applied Mathemathematics
University of Johannesburg
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
1
Linear Equations
2
Linear systems with two and three unknowns
3
Augmented Matrices
4
Elementary Row operations
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
Goals:
To introduce matrices
Develop a way to solve matrices
To interpret the solution (if it exists)
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
Linear Equations
In two dimensions a line is given by:
ax + by = c.
In three dimensions a plane is given by:
ax + by + cz = d.
These are examples of linear equations
Generally, we define a linear equation on the n variables
x1 , x2 , . . . , xn as
a1 x1 + a2 x2 + . . . + an xn = b.
where a1 , a2 , . . . , an and b are constants such that not all the
a’s are zeros.
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
Linear Equations
In two dimensions a line is given by:
ax + by = c.
In three dimensions a plane is given by:
ax + by + cz = d.
These are examples of linear equations
Generally, we define a linear equation on the n variables
x1 , x2 , . . . , xn as
a1 x1 + a2 x2 + . . . + an xn = b.
where a1 , a2 , . . . , an and b are constants such that not all the
a’s are zeros.
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
Linear Equations
In two dimensions a line is given by:
ax + by = c.
In three dimensions a plane is given by:
ax + by + cz = d.
These are examples of linear equations
Generally, we define a linear equation on the n variables
x1 , x2 , . . . , xn as
a1 x1 + a2 x2 + . . . + an xn = b.
where a1 , a2 , . . . , an and b are constants such that not all the
a’s are zeros.
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
Linear Equations
In the special cases where n = 2 or n = 3, we will often use
variables without subscripts and write linear equations as
a1 x + a2 y
= b
a1 x + a2 y + a3 z = b
(a1 , a2 not both 0)
(a1 , a2 , a3 not all 0)
A homogeneous linear equation in the n variables
x1 , x2 , . . . , xn can be written as
a1 x1 + a2 x2 + . . . + an xn = 0.
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
Linear Equations
In the special cases where n = 2 or n = 3, we will often use
variables without subscripts and write linear equations as
a1 x + a2 y
= b
a1 x + a2 y + a3 z = b
(a1 , a2 not both 0)
(a1 , a2 , a3 not all 0)
A homogeneous linear equation in the n variables
x1 , x2 , . . . , xn can be written as
a1 x1 + a2 x2 + . . . + an xn = 0.
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
Linear Equations
In the special cases where n = 2 or n = 3, we will often use
variables without subscripts and write linear equations as
a1 x + a2 y
= b
a1 x + a2 y + a3 z = b
(a1 , a2 not both 0)
(a1 , a2 , a3 not all 0)
A homogeneous linear equation in the n variables
x1 , x2 , . . . , xn can be written as
a1 x1 + a2 x2 + . . . + an xn = 0.
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
Identifying linear equations
Example
Which of the following are linear equations in the variables
x, y , z, w, x1 , x2 , x3 ?
x − 3y + 4z − 17w = 6
Linear
sin x + 7y = 6
Not linear
xy + 6z = 4
Not linear
−3x 2
Not linear
+y =0
5x3 + 4x1 − x2 = 1
Linear
πx1 − x2 = 6
Linear
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
Identifying linear equations
Example
Which of the following are linear equations in the variables
x, y , z, w, x1 , x2 , x3 ?
x − 3y + 4z − 17w = 6
Linear
sin x + 7y = 6
Not linear
xy + 6z = 4
Not linear
−3x 2
Not linear
+y =0
5x3 + 4x1 − x2 = 1
Linear
πx1 − x2 = 6
Linear
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
Identifying linear equations
Example
Which of the following are linear equations in the variables
x, y , z, w, x1 , x2 , x3 ?
x − 3y + 4z − 17w = 6
Linear
sin x + 7y = 6
Not linear
xy + 6z = 4
Not linear
−3x 2
Not linear
+y =0
5x3 + 4x1 − x2 = 1
Linear
πx1 − x2 = 6
Linear
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
Identifying linear equations
Example
Which of the following are linear equations in the variables
x, y , z, w, x1 , x2 , x3 ?
x − 3y + 4z − 17w = 6
Linear
sin x + 7y = 6
Not linear
xy + 6z = 4
Not linear
−3x 2
Not linear
+y =0
5x3 + 4x1 − x2 = 1
Linear
πx1 − x2 = 6
Linear
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
Identifying linear equations
Example
Which of the following are linear equations in the variables
x, y , z, w, x1 , x2 , x3 ?
x − 3y + 4z − 17w = 6
Linear
sin x + 7y = 6
Not linear
xy + 6z = 4
Not linear
−3x 2
Not linear
+y =0
5x3 + 4x1 − x2 = 1
Linear
πx1 − x2 = 6
Linear
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
Identifying linear equations
Example
Which of the following are linear equations in the variables
x, y , z, w, x1 , x2 , x3 ?
x − 3y + 4z − 17w = 6
Linear
sin x + 7y = 6
Not linear
xy + 6z = 4
Not linear
−3x 2
Not linear
+y =0
5x3 + 4x1 − x2 = 1
Linear
πx1 − x2 = 6
Linear
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
Identifying linear equations
Example
Which of the following are linear equations in the variables
x, y , z, w, x1 , x2 , x3 ?
x − 3y + 4z − 17w = 6
Linear
sin x + 7y = 6
Not linear
xy + 6z = 4
Not linear
−3x 2
Not linear
+y =0
5x3 + 4x1 − x2 = 1
Linear
πx1 − x2 = 6
Linear
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
Systems of linear equations
Definition
A system of linear equations or linear system is a finite
collection of linear equations.
Example
x +y −z +w =1
2x − y + 4z = 11
−3x + 4w = −6
The variables are called unknowns.
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
m equations in n unknowns
A linear system of m equations in the n unknowns x1 , x2 , . . . , xn
can be written as
a11 x1
a21 x1
..
.
+
+
a12 x2
a22 x2
..
.
+ ···
+ ···
···
am1 x1 + am2 x2 + · · ·
Dr K
+
+
a1n xn
a2n xn
..
.
=
=
b1
b2
..
.
+ amn xn = bm
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
Solutions to systems of linear equations
Definition
A solution of a linear system in n unknowns x1 , x2 , . . . , xn is a
sequence of n numbers
s1 , s2 , . . . , sn
for which the substitution
x1 = s1 , x2 = s2 ,
...,
xn = sn
makes each equation a true statement.
The solution can be written as (s1 , s2 , · · · , sn ) which is called an
ordered n-tuple.
If n = 2: ordered pair;
if n = 3: ordered triple.
A linear system is called consistent if it has at least one
solution and inconsistent if it has no solution.
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
Solutions to systems of linear equations
Definition
A solution of a linear system in n unknowns x1 , x2 , . . . , xn is a
sequence of n numbers
s1 , s2 , . . . , sn
for which the substitution
x1 = s1 , x2 = s2 ,
...,
xn = sn
makes each equation a true statement.
The solution can be written as (s1 , s2 , · · · , sn ) which is called an
ordered n-tuple.
If n = 2: ordered pair;
if n = 3: ordered triple.
A linear system is called consistent if it has at least one
solution and inconsistent if it has no solution.
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
Solutions to systems of linear equations
Definition
A solution of a linear system in n unknowns x1 , x2 , . . . , xn is a
sequence of n numbers
s1 , s2 , . . . , sn
for which the substitution
x1 = s1 , x2 = s2 ,
...,
xn = sn
makes each equation a true statement.
The solution can be written as (s1 , s2 , · · · , sn ) which is called an
ordered n-tuple.
If n = 2: ordered pair;
if n = 3: ordered triple.
A linear system is called consistent if it has at least one
solution and inconsistent if it has no solution.
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
The solution of Linear systems in two unknowns arise in
connection with intersections of lines.
For example, consider the linear system
a1 x + b1 y
= c1
a2 x + b2 y
= c2
in which the graphs of the equations are lines in the xy-plane.
Each solution (x, y ) of this system corresponds to a point of
intersection of the lines, so there are three possibilities. i.e.
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
The solution of Linear systems in two unknowns arise in
connection with intersections of lines.
For example, consider the linear system
a1 x + b1 y
= c1
a2 x + b2 y
= c2
in which the graphs of the equations are lines in the xy-plane.
Each solution (x, y ) of this system corresponds to a point of
intersection of the lines, so there are three possibilities. i.e.
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
The solution of Linear systems in two unknowns arise in
connection with intersections of lines.
For example, consider the linear system
a1 x + b1 y
= c1
a2 x + b2 y
= c2
in which the graphs of the equations are lines in the xy-plane.
Each solution (x, y ) of this system corresponds to a point of
intersection of the lines, so there are three possibilities. i.e.
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
The lines may be parallel and distinct, in which case there
is no intersection and consequently no solution.
The lines may intersect at only one point, in which case the
system has exactly one solution.
The lines may coincide, in which case there are infinitely
many points of intersection (the points on the common line)
and consequently infinitely many solutions.
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
The lines may be parallel and distinct, in which case there
is no intersection and consequently no solution.
The lines may intersect at only one point, in which case the
system has exactly one solution.
The lines may coincide, in which case there are infinitely
many points of intersection (the points on the common line)
and consequently infinitely many solutions.
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
The lines may be parallel and distinct, in which case there
is no intersection and consequently no solution.
The lines may intersect at only one point, in which case the
system has exactly one solution.
The lines may coincide, in which case there are infinitely
many points of intersection (the points on the common line)
and consequently infinitely many solutions.
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
The same is true for a linear system of three equations in three
unknowns
a1 x + b1 y + c1 z = d1
a2 x + b2 y + c2 z = d2
a3 x + b3 y + c3 z = d3
in which the graphs of the equations are planes.
The solutions of the system, if any, correspond to points where
all three planes intersect, so again there are only three
possibilities?no solutions,one solution, or infinitely many
solutions
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
The same is true for a linear system of three equations in three
unknowns
a1 x + b1 y + c1 z = d1
a2 x + b2 y + c2 z = d2
a3 x + b3 y + c3 z = d3
in which the graphs of the equations are planes.
The solutions of the system, if any, correspond to points where
all three planes intersect, so again there are only three
possibilities?no solutions,one solution, or infinitely many
solutions
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
The same is true for a linear system of three equations in three
unknowns
a1 x + b1 y + c1 z = d1
a2 x + b2 y + c2 z = d2
a3 x + b3 y + c3 z = d3
in which the graphs of the equations are planes.
The solutions of the system, if any, correspond to points where
all three planes intersect, so again there are only three
possibilities?no solutions,one solution, or infinitely many
solutions
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
The same is true for a linear system of three equations in three
unknowns
a1 x + b1 y + c1 z = d1
a2 x + b2 y + c2 z = d2
a3 x + b3 y + c3 z = d3
in which the graphs of the equations are planes.
The solutions of the system, if any, correspond to points where
all three planes intersect, so again there are only three
possibilities?no solutions,one solution, or infinitely many
solutions
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
The same is true for a linear system of three equations in three
unknowns
a1 x + b1 y + c1 z = d1
a2 x + b2 y + c2 z = d2
a3 x + b3 y + c3 z = d3
in which the graphs of the equations are planes.
The solutions of the system, if any, correspond to points where
all three planes intersect, so again there are only three
possibilities?no solutions,one solution, or infinitely many
solutions
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
The same is true for a linear system of three equations in three
unknowns
a1 x + b1 y + c1 z = d1
a2 x + b2 y + c2 z = d2
a3 x + b3 y + c3 z = d3
in which the graphs of the equations are planes.
The solutions of the system, if any, correspond to points where
all three planes intersect, so again there are only three
possibilities?no solutions,one solution, or infinitely many
solutions
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
Example of linear system with one solution
Example
Solve the linear systems:
x +y =1
2x − y = 11
x =4
y = −3
or (x, y ) = (4, −3).
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
Example of linear system with no solution
Example
Solve the linear systems:
x +y =4
3x + 3y = 6
no solution
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
Example of linear system with infinitely many solution
Example
Solve the linear systems:
4x − 2y = 1
16x − 8y = 4
infinitely many solutions
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
Augmented Matrices
Definition
The linear system
a11 x1
a21 x1
..
.
+
+
a12 x2
a22 x2
..
.
+ ···
+ ···
···
am1 x1 + am2 x2 + · · ·
+
+
a1n xn
a2n xn
..
.
=
=
b1
b2
..
.
+ amn xn = bm
can be abbreviated to the augmented matrix


a11 a12 · · ·
a1n
b1
 a21 a22 · · · a2n xn b2 


 ..
..
..
.. 
 .
.
···
.
. 
am1 am2 · · ·
Dr K
amn
bm
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
Augmented matrices
Example
The system of linear equations
x1 + x2 + 2x3 = 9
2x1 + 4x2 − 3x3 = 1
3x1 + 6x2 − 5x3

1 1

has the augmented matrix 2 4
3 6
Dr K
=0

2 9
−3 1 
−5 0
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
The basic method for solving a linear system is to perform
algebraic operations on the system.
These operations do not alter the solution set and that produce
a succession of increasingly simpler systems, until a point is
reached where it can be ascertained whether the system is
consistent, and if so, what its solutions are.
Typically, the algebraic operations are:
Multiply an equation through by a nonzero constant.
Interchange two equations.
Add a constant times one equation to another.
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
The basic method for solving a linear system is to perform
algebraic operations on the system.
These operations do not alter the solution set and that produce
a succession of increasingly simpler systems, until a point is
reached where it can be ascertained whether the system is
consistent, and if so, what its solutions are.
Typically, the algebraic operations are:
Multiply an equation through by a nonzero constant.
Interchange two equations.
Add a constant times one equation to another.
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
The basic method for solving a linear system is to perform
algebraic operations on the system.
These operations do not alter the solution set and that produce
a succession of increasingly simpler systems, until a point is
reached where it can be ascertained whether the system is
consistent, and if so, what its solutions are.
Typically, the algebraic operations are:
Multiply an equation through by a nonzero constant.
Interchange two equations.
Add a constant times one equation to another.
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
The basic method for solving a linear system is to perform
algebraic operations on the system.
These operations do not alter the solution set and that produce
a succession of increasingly simpler systems, until a point is
reached where it can be ascertained whether the system is
consistent, and if so, what its solutions are.
Typically, the algebraic operations are:
Multiply an equation through by a nonzero constant.
Interchange two equations.
Add a constant times one equation to another.
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
The basic method for solving a linear system is to perform
algebraic operations on the system.
These operations do not alter the solution set and that produce
a succession of increasingly simpler systems, until a point is
reached where it can be ascertained whether the system is
consistent, and if so, what its solutions are.
Typically, the algebraic operations are:
Multiply an equation through by a nonzero constant.
Interchange two equations.
Add a constant times one equation to another.
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
Since the rows (horizontal lines) of an augmented matrix
correspond to the equations in the associated system,these
three operations correspond to the following operations on the
rows of the augmented matrix:
Multiply a row through by a nonzero constant.
Interchange two rows.
Add a constant times one row to another.
These are called elementary row operations on a matrix.
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
Since the rows (horizontal lines) of an augmented matrix
correspond to the equations in the associated system,these
three operations correspond to the following operations on the
rows of the augmented matrix:
Multiply a row through by a nonzero constant.
Interchange two rows.
Add a constant times one row to another.
These are called elementary row operations on a matrix.
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
Since the rows (horizontal lines) of an augmented matrix
correspond to the equations in the associated system,these
three operations correspond to the following operations on the
rows of the augmented matrix:
Multiply a row through by a nonzero constant.
Interchange two rows.
Add a constant times one row to another.
These are called elementary row operations on a matrix.
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
Elementary row operations
Definition
The following elementary row operations can be performed
on the rows (horizontal lines) of an augmented matrix:
1
Multiply a row by a nonzero constant.
cRi
(Example: −3R2 )
2
Interchange two rows.
Ri ↔ Rj
(Example: R1 ↔ R3 )
3
Add a constant times one row to another row.
Ri + kRj
(Example: R2 + 4R1 )
Note that:
Ri denotes the ith row.
The row you are changing must be written first.
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
Example
Use row operations to solve the system
x + y + 2z = 9
2x + 4y − 3z = 1
3x + 6y − 5z = 0
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
Solution:
Augmented

1 1 2
2 4 −3
3 6 −5
x + y + 2z = 9
2x + 4y − 3z = 1
3x + 6y − 5z = 0


1 1 2
9
R2 −2R1
−→ 0 2 −7 −17 
3 6 −5
0


1 1
2
9

−→0 1 − 72 − 17
2
0 3 −11 −27
1
R
2 2
matrix

9
1 
0


1 1
2
9
R3 −3R1
−→ 0 2 −7 −17 
0 3 −11 −27

1 1 2
R3 −3R2

0 1 − 72
−→
0 0 − 12
Dr K
9
− 17
2
− 23


Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
Solution:
Augmented

1 1 2
2 4 −3
3 6 −5
x + y + 2z = 9
2x + 4y − 3z = 1
3x + 6y − 5z = 0


1 1 2
9
R2 −2R1
−→ 0 2 −7 −17 
3 6 −5
0


1 1
2
9

−→0 1 − 72 − 17
2
0 3 −11 −27
1
R
2 2
matrix

9
1 
0


1 1
2
9
R3 −3R1
−→ 0 2 −7 −17 
0 3 −11 −27

1 1 2
R3 −3R2

0 1 − 72
−→
0 0 − 12
Dr K
9
− 17
2
− 23


Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
Solution:
Augmented

1 1 2
2 4 −3
3 6 −5
x + y + 2z = 9
2x + 4y − 3z = 1
3x + 6y − 5z = 0


1 1 2
9
R2 −2R1
−→ 0 2 −7 −17 
3 6 −5
0


1 1
2
9

−→0 1 − 72 − 17
2
0 3 −11 −27
1
R
2 2
matrix

9
1 
0


1 1
2
9
R3 −3R1
−→ 0 2 −7 −17 
0 3 −11 −27

1 1 2
R3 −3R2

0 1 − 72
−→
0 0 − 12
Dr K
9
− 17
2
− 23


Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
Solution:
Augmented

1 1 2
2 4 −3
3 6 −5
x + y + 2z = 9
2x + 4y − 3z = 1
3x + 6y − 5z = 0


1 1 2
9
R2 −2R1
−→ 0 2 −7 −17 
3 6 −5
0


1 1
2
9

−→0 1 − 72 − 17
2
0 3 −11 −27
1
R
2 2
matrix

9
1 
0


1 1
2
9
R3 −3R1
−→ 0 2 −7 −17 
0 3 −11 −27

1 1 2
R3 −3R2

0 1 − 72
−→
0 0 − 12
Dr K
9
− 17
2
− 23


Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
Solution:
Augmented

1 1 2
2 4 −3
3 6 −5
x + y + 2z = 9
2x + 4y − 3z = 1
3x + 6y − 5z = 0


1 1 2
9
R2 −2R1
−→ 0 2 −7 −17 
3 6 −5
0


1 1
2
9

−→0 1 − 72 − 17
2
0 3 −11 −27
1
R
2 2
matrix

9
1 
0


1 1
2
9
R3 −3R1
−→ 0 2 −7 −17 
0 3 −11 −27

1 1 2
R3 −3R2

0 1 − 72
−→
0 0 − 12
Dr K
9
− 17
2
− 23


Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
Solution:
Augmented

1 1 2
2 4 −3
3 6 −5
x + y + 2z = 9
2x + 4y − 3z = 1
3x + 6y − 5z = 0


1 1 2
9
R2 −2R1
−→ 0 2 −7 −17 
3 6 −5
0


1 1
2
9

−→0 1 − 72 − 17
2
0 3 −11 −27
1
R
2 2
matrix

9
1 
0


1 1
2
9
R3 −3R1
−→ 0 2 −7 −17 
0 3 −11 −27

1 1 2
R3 −3R2

0 1 − 72
−→
0 0 − 12
Dr K
9
− 17
2
− 23


Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
Example continued

1 1 2
0 1 − 7
2
0 0 − 21
9
− 17
2
− 32

1 0 11
2
R1 −R2
−→ 0 1 − 72
0 0 1


35
2
− 17
2

1 1 2
−2R3

−→ 0 1 − 72
0 0 1

9
− 17
2
3


R1 − 11 R3
2


1 0 0 1
−→ 0 1 0 2 
0 0 1 3
R2 + 72 R3

3
The solution x = 1, y = 2, z = 3 is now evident.
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
Example continued

1 1 2
0 1 − 7
2
0 0 − 21
9
− 17
2
− 32

1 0 11
2
R1 −R2
−→ 0 1 − 72
0 0 1


35
2
− 17
2

1 1 2
−2R3

−→ 0 1 − 72
0 0 1

9
− 17
2
3


R1 − 11 R3
2


1 0 0 1
−→ 0 1 0 2 
0 0 1 3
R2 + 72 R3

3
The solution x = 1, y = 2, z = 3 is now evident.
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
Example continued

1 1 2
0 1 − 7
2
0 0 − 21
9
− 17
2
− 32

1 0 11
2
R1 −R2
−→ 0 1 − 72
0 0 1


35
2
− 17
2

1 1 2
−2R3

−→ 0 1 − 72
0 0 1

9
− 17
2
3


R1 − 11 R3
2


1 0 0 1
−→ 0 1 0 2 
0 0 1 3
R2 + 72 R3

3
The solution x = 1, y = 2, z = 3 is now evident.
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
Example continued

1 1 2
0 1 − 7
2
0 0 − 21
9
− 17
2
− 32

1 0 11
2
R1 −R2
−→ 0 1 − 72
0 0 1


35
2
− 17
2

1 1 2
−2R3

−→ 0 1 − 72
0 0 1

9
− 17
2
3


R1 − 11 R3
2


1 0 0 1
−→ 0 1 0 2 
0 0 1 3
R2 + 72 R3

3
The solution x = 1, y = 2, z = 3 is now evident.
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
Example continued

1 1 2
0 1 − 7
2
0 0 − 21
9
− 17
2
− 32

1 0 11
2
R1 −R2
−→ 0 1 − 72
0 0 1


35
2
− 17
2

1 1 2
−2R3

−→ 0 1 − 72
0 0 1

9
− 17
2
3


R1 − 11 R3
2


1 0 0 1
−→ 0 1 0 2 
0 0 1 3
R2 + 72 R3

3
The solution x = 1, y = 2, z = 3 is now evident.
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
Example continued

1 1 2
0 1 − 7
2
0 0 − 21
9
− 17
2
− 32

1 0 11
2
R1 −R2
−→ 0 1 − 72
0 0 1


35
2
− 17
2

1 1 2
−2R3

−→ 0 1 − 72
0 0 1

9
− 17
2
3


R1 − 11 R3
2


1 0 0 1
−→ 0 1 0 2 
0 0 1 3
R2 + 72 R3

3
The solution x = 1, y = 2, z = 3 is now evident.
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
Example continued

1 1 2
0 1 − 7
2
0 0 − 21
9
− 17
2
− 32

1 0 11
2
R1 −R2
−→ 0 1 − 72
0 0 1


35
2
− 17
2

1 1 2
−2R3

−→ 0 1 − 72
0 0 1

9
− 17
2
3


R1 − 11 R3
2


1 0 0 1
−→ 0 1 0 2 
0 0 1 3
R2 + 72 R3

3
The solution x = 1, y = 2, z = 3 is now evident.
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
In conclusion: Do you know these concepts?
Linear Equation
Consistent linear system
Homogenous linear
equation
Inconsistent linear
system
System of linear
equation
Augmented matrix
Elementary row
operations
Solution of a linear
system
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
Can you?
Determine whether a given equation is linear.
Determine whether a given n-tuple is a solution of a linear
system
Find the augmented matrix of a linear system
Find the linear system corresponding to a given
augmented matrix
Perform elementary row operations on a linear system and
on its corresponding augmented matrix
Determine whether a linear system is consistent or
inconsistent
Find the solution to a consistent linear system
Dr K
Introduction to systems of linear equations
Linear Equations
Linear systems with two and three unknowns
Augmented Matrices
Elementary Row operations
Exercise:
Solve the following linear systems
(a)
x + 3y + z = 5
2x + 7y + 2z = 4
x + 2y + 2z = 3
(b)
3y − z = 4
x + 3y + z = 1
x + 2z = 4
(c)
2x + 4y − 6z = 2
−x − 2y + 3z = −1
x +y +z = 3
(a) (x, y , z) = (31, −6, −8)
(b) Inconsistent.
(c)(x, y , z) = (5, −2, 0) + t(−5, 4, 1)
Dr K
Introduction to systems of linear equations