Systems of Linear Equations
(Part 1)
MODULE 1
MS. KENNETH N. TERCERO
Intended Learning Outcomes
1.
Identify a linear equation and system of linear equation classification based on its
solution set.
2.
Solve system of linear equations algebraically.
3.
Apply Gaussian Elimination and Gauss-Jordan Elimination in solving systems of linear
equations.
4.
Reflect on how systems of linear equation can help solve simple real life situations.
5.
Investigate the difference between Gaussian elimination and Gauss Jordan Elimination
method.
6.
Manipulate MATLAB to solve simple arithmetic problem.
7.
Create a practical/ real life solution that will require the use of the system of linear
equations.
Linear Equation

Example:
Determine if the following are linear or non- linear equation

Parametric Representation

To represent the infinite number of solutions of a linear equation, it is convenient to introduce a
third variable t called a parameter, where t is any real number.

Particular solutions can be obtained by assigning values to the parameter t

There are n-m free variables, variables can take on any value.

m =number of row ;n= number of variable
System of Linear Equations (SLE)

Solution of System of Linear Equation

Number of Solutions of a system of Two
Linear Equations in Two Variables

One Solution (Consistent System)
- The graph of a system is two intersecting lines.
𝑎
𝑏
- 1≠ 1
𝑎2

No Solution (Inconsistent System)
- The graph of a system is two parallel lines
𝑎1
𝑏1
𝑐1
- = ≠
𝑎2

𝑏2
𝑏2
𝑐2
Infinitely Many Solution (Consistent System)
- The graph of a system is two coincides lines.
𝑎
𝑏
𝑐
- 1= 1= 1
𝑎2
𝑏2
𝑐2
Finding the Solution Set of SLE
METHOD OF ELIMINATION:
that is, we eliminate some variables by adding a multiple
of one equation to another equation. Elimination merely
amounts to the development of a new linear system that
is equivalent to the original system, but is much simpler to
solve.
Graph of System of Linear Equation
Systems of Linear Equations
(Part 2)
MODULE 1
MR. KENNETH N. TERCERO
Intended Learning Outcomes
1.
Perform row operations on a matrix.
2.
Solve system of linear equations by Gaussian elimination and GaussJordan Elimination.
Matrix
Is a rectangular array of numbers denoted by:
A=
a11
a 21
a 31
a 41
a12
a 22
a 32
a m1
....
....
....
....
a1n
:
:
a mn
The dimension /order of matrix mxn where m is the rows , n is the columns.
Name: B
Dimension: 2x3
b23 = 2
b32 = No entry
Linear Systems
Coefficient matrix
Variable
Matrix
Constant
matrix
Linear System

Solving System of Linear Equation
Gaussian Elimination Method

Gaussian Elimination is a method for solving
systems of linear equations with several
unknown variables. It works by bringing the
matrix representing the equations into row
echelon form and resolving the unknown
variables by back-substitution.
Gauss Jordan Elimination,

More commonly known as the elimination
method, is a process to solve systems of
linear equations with several unknown
variables. It works by bringing the equations
that contain the unknown variables into
reduced row echelon form. It is an extension
of Gaussian Elimination which brings the
equations into row-echelon form.
Solving System of Linear Equation by Gaussian Elimination & GaussJordan Elimination
Step 1: Create an Augmented Matrix
The matrix A is called the coefficient matrix of the linear system and the matrix obtained
by adjoining column b to A, is called the augmented matrix of the linear system. The
augmented matrix of System of linear equation is written as [A : b ]
Step 2: Transform the Augmented Matrix to REF or RREF
ROW- ECHELON FORM AND REDUCED ROW ECHELON FORM(RREF)
•
A matrix form used when solving linear systems of equations. A matrix is in rowechelon form if it satisfies the following conditions:
1.
The first non zero number in each row ( from left to right) is 1. This is called the
leading entry.
2.
The leading entry in each row is to the right of leading entry in the row
immediately above it.
3.
All rows consisting entirely of zeros are at the bottom of the matrix.
4.
A matrix is in reduced row echelon form if it is in row echelon form and also
satisfies the condition:
5.
Every number above and below each leading entry is a zero.
Example
Matrix in Row- Echelon Form (REF) :
1 𝑎
0 1
0 0
𝑏
𝑐
1
𝑑
𝑒
𝑓
1 0
0 1
0 0
0
0
1
𝑑
𝑒
𝑓
Matrix in Reduced Row- Echelon Form
(RREF) :
To transform a matrix to row echelon or reduced row echelon form we need to perform row
operations.
ROW OPERATIONS
Row operations are ways to change matrices.
There are three types of row operations: These row operations are used in a number
of ways including solving linear equations and finding inverses.
1. Row switching, that is interchanging two rows of a matrix,
2. Row multiplication, multiplying all entries of a row by a non-zero constant
3. Row addition which means adding a multiple of a row to another row
Gaussian Elimination & Gauss-Jordan
Elimination Method
Solution:
𝑅𝑇𝐶 = −𝐸𝑅𝑇𝐶 𝑅𝐿𝐸 + 𝑅𝑇𝐶
𝑅𝑇𝐶 = −𝐸𝑅𝑇𝐶 𝑅𝐿𝐸 + 𝑅𝑇𝐶
System of Linear Equation
With No Solution
System of Linear Equation
with Infinite Solution
Solution:
R1=1/2 R1
1 2 − 1 0
3 5 0 1


R2 =-1R2
1 2 − 1 0 
0 1 − 3 − 1


R2= -3R1 + R2
1 2 − 1 0
0 − 1 3 1 


R1=-2R2 + R1
2
1 0 5
0 1 − 3 − 1


REMARKS:
After doing the row operation on the augmented matrix and the resulting matrix is as
follow:
No solution / inconsistent
Infinite solution/consistent – Need to proceed to parametric representation
Homogeneous System of Linear Equation
Example :
Homogeneous System of Linear Equation
“ Ask & Learn “

Type in the chat box your questions & clarification or you may use your
microphone.
PLEASE TAKE THE
QUIZ 1 (PRELIM )- SYSTEM
OF LINEAR EQUATION
ON THE MODULE 1 IN
CANVAS.
MATLAB
(MATRIX LABORATORY)
USING THE COMMAND WINDOW, PLEASE FIND THE MATLAB SYNTAX AND OUTPUT OF THE
FOLLOWING
Given Expressions
MATLAB SYNTAX
MATLAB OUTPUT
5+2
7
2.) 6 + 3
5/6 + 2/3
1.5000
3.) (2 + 5)3
(2 + 5)^3
343
4.) 2(3)2
2*(3)^2
18
5.) 2 + 6
sqrt(2+6)
2.8284
1.) 5 + 2
5
2
PLEASE ANSWER THE
SIMULATION ACTIVITY 1- BASIC
FUNCTIONALITY OF MATLAB AS A
MATHEMATICAL TOOL
ON MODULE 1 IN CANVAS