PPT: System of Linear Equations

# PPT: System of Linear Equations | Engineering Mathematics - Civil Engineering (CE) PDF Download

``` Page 1

Linear System
? Linear equation: ax
1
+ bx
2
= c
where, a, b, c = constants                 x
1,
x
2
= variables
? A linear system consists of more than one linear equations.
? For example:- a
1
x
1
+ b
1
x
2
+ c
1
x
3
= k
1
……………….(1)
a
2
x
1
+ b
2
x
2
+ c
2
x
2
= k
2
……………….(2)
a
3
x
1
+ b
3
x
2
+ c
3
x
3
= k
3
……………….(3)
? To find a solution of this system means to find the value/values of x
1,
x
2
, x
3
which satisfies all
the equations in the linear system.
? A linear system can have a unique solution, more than one solutions or no solution.
Page 2

Linear System
? Linear equation: ax
1
+ bx
2
= c
where, a, b, c = constants                 x
1,
x
2
= variables
? A linear system consists of more than one linear equations.
? For example:- a
1
x
1
+ b
1
x
2
+ c
1
x
3
= k
1
……………….(1)
a
2
x
1
+ b
2
x
2
+ c
2
x
2
= k
2
……………….(2)
a
3
x
1
+ b
3
x
2
+ c
3
x
3
= k
3
……………….(3)
? To find a solution of this system means to find the value/values of x
1,
x
2
, x
3
which satisfies all
the equations in the linear system.
? A linear system can have a unique solution, more than one solutions or no solution.
Linear System
? Rewriting the above equation in form of matrix
AX = B
A =
?? 1
?? 1
?? 1
?? 2
?? 2
?? 2
?? 3
?? 3
?? 3
, X =
?? 1
?? 2
?? 3
, B =
?? 1
?? 2
?? 3
? The matrix A is called the coefficient matrix and the block matrix [A B] , is the augmented
matrix of the linear system. Type eq uati on h er e.
Augmented matrix :
?? 1
?? 1
?? 1
?? 1
?? 2
?? 2
?? 2
?? 2
?? 3
?? 3
?? 3
?? 3
? If all the elements of B are zero then linear system is called Homogeneous otherwise Non-
Homogeneous.
Page 3

Linear System
? Linear equation: ax
1
+ bx
2
= c
where, a, b, c = constants                 x
1,
x
2
= variables
? A linear system consists of more than one linear equations.
? For example:- a
1
x
1
+ b
1
x
2
+ c
1
x
3
= k
1
……………….(1)
a
2
x
1
+ b
2
x
2
+ c
2
x
2
= k
2
……………….(2)
a
3
x
1
+ b
3
x
2
+ c
3
x
3
= k
3
……………….(3)
? To find a solution of this system means to find the value/values of x
1,
x
2
, x
3
which satisfies all
the equations in the linear system.
? A linear system can have a unique solution, more than one solutions or no solution.
Linear System
? Rewriting the above equation in form of matrix
AX = B
A =
?? 1
?? 1
?? 1
?? 2
?? 2
?? 2
?? 3
?? 3
?? 3
, X =
?? 1
?? 2
?? 3
, B =
?? 1
?? 2
?? 3
? The matrix A is called the coefficient matrix and the block matrix [A B] , is the augmented
matrix of the linear system. Type eq uati on h er e.
Augmented matrix :
?? 1
?? 1
?? 1
?? 1
?? 2
?? 2
?? 2
?? 2
?? 3
?? 3
?? 3
?? 3
? If all the elements of B are zero then linear system is called Homogeneous otherwise Non-
Homogeneous.
Row Reduced Echelon Form of a
Matrix
? A matrix C is said to be in the row reduced form if
1. The first non-zero entry in each row of C is 1
2. The column containing this 1 has all its other entries zero
e.g. A =
1 1 3
0 3 2
0 0 1
B =
1 0 1 - 2
0 1 2 3
0 0 4 1
0 0 - 1 - 2
C =
1 - 1
0 2
Page 4

Linear System
? Linear equation: ax
1
+ bx
2
= c
where, a, b, c = constants                 x
1,
x
2
= variables
? A linear system consists of more than one linear equations.
? For example:- a
1
x
1
+ b
1
x
2
+ c
1
x
3
= k
1
……………….(1)
a
2
x
1
+ b
2
x
2
+ c
2
x
2
= k
2
……………….(2)
a
3
x
1
+ b
3
x
2
+ c
3
x
3
= k
3
……………….(3)
? To find a solution of this system means to find the value/values of x
1,
x
2
, x
3
which satisfies all
the equations in the linear system.
? A linear system can have a unique solution, more than one solutions or no solution.
Linear System
? Rewriting the above equation in form of matrix
AX = B
A =
?? 1
?? 1
?? 1
?? 2
?? 2
?? 2
?? 3
?? 3
?? 3
, X =
?? 1
?? 2
?? 3
, B =
?? 1
?? 2
?? 3
? The matrix A is called the coefficient matrix and the block matrix [A B] , is the augmented
matrix of the linear system. Type eq uati on h er e.
Augmented matrix :
?? 1
?? 1
?? 1
?? 1
?? 2
?? 2
?? 2
?? 2
?? 3
?? 3
?? 3
?? 3
? If all the elements of B are zero then linear system is called Homogeneous otherwise Non-
Homogeneous.
Row Reduced Echelon Form of a
Matrix
? A matrix C is said to be in the row reduced form if
1. The first non-zero entry in each row of C is 1
2. The column containing this 1 has all its other entries zero
e.g. A =
1 1 3
0 3 2
0 0 1
B =
1 0 1 - 2
0 1 2 3
0 0 4 1
0 0 - 1 - 2
C =
1 - 1
0 2
Gauss Elimination Method
? Gaussian elimination is a method of solving a linear system AX = B (consisting of m
equations in n unknowns) by bringing the augmented matrix
[A B] =
?? 11 ??1 2 ? ??1 ?? ??1
??2 1 ??2 2 ? ??2 ?? ??2
? ? ? ? ?
????1 ????2 ? ?????? ????
? to an upper triangular form (or reduced row echelon form)
?? 11 ?? 12 ? ?? 1?? ?? 1
0 ?? 22 ? ?? 2?? ?? 2
? ? ? ? ?
0 0 ? ?? ???? ?? ?? ? This elimination process is also called the forward elimination method.
Page 5

Linear System
? Linear equation: ax
1
+ bx
2
= c
where, a, b, c = constants                 x
1,
x
2
= variables
? A linear system consists of more than one linear equations.
? For example:- a
1
x
1
+ b
1
x
2
+ c
1
x
3
= k
1
……………….(1)
a
2
x
1
+ b
2
x
2
+ c
2
x
2
= k
2
……………….(2)
a
3
x
1
+ b
3
x
2
+ c
3
x
3
= k
3
……………….(3)
? To find a solution of this system means to find the value/values of x
1,
x
2
, x
3
which satisfies all
the equations in the linear system.
? A linear system can have a unique solution, more than one solutions or no solution.
Linear System
? Rewriting the above equation in form of matrix
AX = B
A =
?? 1
?? 1
?? 1
?? 2
?? 2
?? 2
?? 3
?? 3
?? 3
, X =
?? 1
?? 2
?? 3
, B =
?? 1
?? 2
?? 3
? The matrix A is called the coefficient matrix and the block matrix [A B] , is the augmented
matrix of the linear system. Type eq uati on h er e.
Augmented matrix :
?? 1
?? 1
?? 1
?? 1
?? 2
?? 2
?? 2
?? 2
?? 3
?? 3
?? 3
?? 3
? If all the elements of B are zero then linear system is called Homogeneous otherwise Non-
Homogeneous.
Row Reduced Echelon Form of a
Matrix
? A matrix C is said to be in the row reduced form if
1. The first non-zero entry in each row of C is 1
2. The column containing this 1 has all its other entries zero
e.g. A =
1 1 3
0 3 2
0 0 1
B =
1 0 1 - 2
0 1 2 3
0 0 4 1
0 0 - 1 - 2
C =
1 - 1
0 2
Gauss Elimination Method
? Gaussian elimination is a method of solving a linear system AX = B (consisting of m
equations in n unknowns) by bringing the augmented matrix
[A B] =
?? 11 ??1 2 ? ??1 ?? ??1
??2 1 ??2 2 ? ??2 ?? ??2
? ? ? ? ?
????1 ????2 ? ?????? ????
? to an upper triangular form (or reduced row echelon form)
?? 11 ?? 12 ? ?? 1?? ?? 1
0 ?? 22 ? ?? 2?? ?? 2
? ? ? ? ?
0 0 ? ?? ???? ?? ?? ? This elimination process is also called the forward elimination method.
Gauss Elimination Method
E.X. 1       y + z = 2
2x + 3z = 5
x + y + z = 5
Solution:- Augmented matrix =
0 1 1 2
2 0 3 5
1 1 1 3
? Interchange 1
st
and 2
nd
equation
2 0 3 5
0 1 1 2
1 1 1 3
```

## Engineering Mathematics

65 videos|120 docs|94 tests

## FAQs on PPT: System of Linear Equations - Engineering Mathematics - Civil Engineering (CE)

 1. What is a system of linear equations?
Ans. A system of linear equations is a set of two or more equations with the same variables. The solution to the system is the values of the variables that satisfy all the equations simultaneously.
 2. How can I solve a system of linear equations?
Ans. There are several methods to solve a system of linear equations, including substitution, elimination, and matrix methods. The choice of method depends on the complexity of the system and personal preference.
 3. Can a system of linear equations have no solution?
Ans. Yes, a system of linear equations can have no solution. This occurs when the equations are inconsistent, meaning there is no set of values that satisfies all the equations simultaneously. Geometrically, this represents parallel lines that never intersect.
 4. Is it possible for a system of linear equations to have infinitely many solutions?
Ans. Yes, a system of linear equations can have infinitely many solutions. This happens when the equations are dependent, meaning one equation can be obtained by combining the others. Geometrically, this corresponds to overlapping lines or planes.
 5. Can a system of linear equations have a unique solution?
Ans. Yes, a system of linear equations can have a unique solution. This occurs when the equations are independent, meaning none of the equations can be derived from others. Geometrically, this represents intersecting lines or planes at a single point.

## Engineering Mathematics

65 videos|120 docs|94 tests

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