SM015 - 4.4 (Notes Solutions)

4.4 System of Linear Equations with 3 Variables

Friday, July 10, 2020 3:14 PM

Learning Outcomes:

At the end of the lesson, students should be able to

(a) Write a system of linear equations in the form of

(b) Solve the unique solution of using:

(i) Inverse Matrix;
(ii) Elimination Method.
(iii) Cramer's Rule

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System of Linear Equations

 Consider the system of linear equations with three variables , and .

The above system of linear equation can be written as a single matrix
equation as below:

If ,

then the system of linear equations can be written as

 Then, the system of linear equation can be solved by whichever one of

the following method: Remarks:
(a) Inverse Method There are 3 type of solutions for the matrix
(b) Elimination Method equation :

(c) Cramer's Rule (i) Unique solution
(ii) Infinite solution

(iii) Inconsistent / no solution

• Unique solution

• Infinite solution or NO solution

4.4 System of Linear Equations with 3 Variables Page 1

Inverse Method

Monday, May 04, 2020 4:10 PM

 If the number of equations in a system equals to the number of variables
and the coefficient matrix has an inverse, then the system will always
have a unique solution that can be found by using the inverse of the
coefficient matrix.

 The solution of a matrix equation is given by

where square matrix with all the coefficient & has an inverse ,
is a matrix with n variables and
is a matrix with n constants.

B is a

Remarks:
can be obtained through either one of the following method:
(a) adjoint method
(b) ERO
(c) properties of inverse matrix
(d) given by the question itself

Example 1

Solve the system of linear equation by using the inverse matrix method.

Solution:

Since , then there are infinite solutions or no solution.

4.4 System of Linear Equations with 3 Variables Page 2

Example 2 (Part I)

Friday, May 29, 2020 10:16 PM

Given and

Find and hence deduce the matrix .

Solution:

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Example 2 (Part II) when .

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Previously, it is found that

Express the following system in the matrix form

and hence solve the linear system.
Solution:

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Example 3 (Part I) by using adjoint method.

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Find the inverse matrix of

Solution:

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Example 3 (Part II) when

Friday, May 29, 2020 10:16 PM

Previously, it is found that

Hence, solve the system of linear equations by using the inverse
matrix method.

Solution:

,,

4.4 System of Linear Equations with 3 Variables Page 6

Elimination Method

Monday, May 04, 2020 4:10 PM

 Elimination is a method of solving system of linear equations by
using ERO.

 2 type of Elimination Methods:
(a) Gaussian Elimination method
(b) Gauss-Jordan Elimination method

 Procedure:

(I) Convert the system of linear equation into matrix equation

.

(II) Form an augmented matrix in the form of .

(III) Reduce the augmented matrix by using ERO.

▪ Gaussian Elimination method

Then, extract the reduced linear equations from
where is an upper triangular matrix & solve the
simultatenous equations to obtained solution.

▪ Gauss-Jordan Elimination method

Then, extract the solution from matrix .
Remarks:

4.4 System of Linear Equations with 3 Variables Page 7

Gaussian Elimination VS Gauss-Jordan Elimination

Monday, May 04, 2020 4:08 PM

By Elimination method, solve the following system of linear equations:
Solution:

Gaussian Elimination method Gauss-Jordan Elimination method

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Example 4 (Gaussian Elimination )

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Solve the following system of linear equations using Gaussian
Elimination method.
Solution:

Hence, [By Gaussian Elimination method]

,,

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Example 4 (Gauss-Jordan Elimination)

Friday, May 29, 2020 10:16 PM

Solve the following system of linear equations using Gauss-Jordan
Elimination method.
Solution:

,,

4.4 System of Linear Equations with 3 Variables Page 10

Cramer's Rule

Monday, May 04, 2020 4:10 PM

 Consider the system of linear equations with three variables , and .

The above system of linear equation can be written as a single matrix

equation as below:

By Cramer's Rule, the solution of this system of linear equations is

4.4 System of Linear Equations with 3 Variables Page 11

Example 1

Friday, May 29, 2020 10:16 PM

By using Cramer's Rule, solve the following system of linear equations.
Solution:

4.4 System of Linear Equations with 3 Variables Page 12

Example 2(a)

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The following table shows the price (RM) per type of 0.5 kg cakes sold at
the shops P , Q and R together with the total expenditure if a customer
buys a number of each type of cake from the listed shops.

Cake types Banana Chocolate Total
Shops Vanilla Expenditure

(RM)

Let the number of banana, chocolate and vanilla cakes bought from each
shop be , and respectively.

(a) Write the matrix equation using the above information.
Solution:

System of linear equations

Matrix Equation

4.4 System of Linear Equations with 3 Variables Page 13

Example 2(b)

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Previously,

(b) By using Cramer's Rule, determine the values of , and .
Solution:

4.4 System of Linear Equations with 3 Variables Page 14