Higher Engineering Mathematics, Sixth Edition

The solution of simultaneous equations by matrices and determinants 249

Use the Gaussian elimination method to solve for

I 1 ,I 2 andI 3.

(This is the same example as Problem 6 on page 243,
and a comparison of methods may be made)

Following the above procedure:

1. 2I 1 + 3 I 2 − 4 I 3 = 26 ( 1 )
   Equation( 2 )−

1

2

×equation (1) gives:

0 − 6. 5 I 2 −I 3 =− 100 ( 2 ′)

Equation( 3 )−

− 7

2

×equation (1) gives:

0 + 12. 5 I 2 − 8 I 3 = 103 ( 3 ′)

2. 2I 1 + 3 I 2 − 4 I 3 = 26 ( 1 )
   0 − 6. 5 I 2 −I 3 =− 100 ( 2 ′)

Equation( 3 ′)−

12. 5

− 6. 5

×equation (2′)gives:

0 + 0 − 9. 923 I 3 =− 89. 308 ( 3 ′′)

3. From equation (3′′),

I 3 =

− 89. 308

− 9. 923

=9mA,

from equation( 2 ′),− 6. 5 I 2 − 9 =−100,

from which,I 2 =

− 100 + 9

− 6. 5

=14mA

and from equation (1), 2I 1 + 3 ( 14 )− 4 ( 9 )= 26 ,

from which,I 1 =

26 − 42 + 36

2

=

20

2

=10mA

Now try the following exercise

Exercise 101 Further problems on solving

simultaneous equations using Gaussian

elimination

1. In a mass-spring-damper system, the acceler-
   ationx ̈m/s^2 , velocityx ̇m/s and displacement
   xm are related by the following simultaneous
   equations:

6. 2 x ̈+ 7. 9 x ̇+ 12. 6 x= 18. 0

7. 5 x ̈+ 4. 8 x ̇+ 4. 8 x= 6. 39

13. 0 x ̈+ 3. 5 x ̇− 13. 0 x=− 17. 4

By using Gaussian elimination, determine the

acceleration, velocity and displacement for the

system, correct to 2 decimal places.

[x ̈=− 0. 30 ,x ̇= 0. 60 ,x= 1 .20]

2. The tensions,T 1 ,T 2 andT 3 in a simple frame-
   work are given by the equations:

5 T 1 + 5 T 2 + 5 T 3 = 7. 0

T 1 + 2 T 2 + 4 T 3 = 2. 4

4 T 1 + 2 T 2 = 4. 0

DetermineT 1 ,T 2 andT 3 using Gaussian elim-

ination.

[T 1 = 0. 8 ,T 2 = 0. 4 ,T 3 = 0 .2]

3. Repeat problems 3, 4, 5, 7 and 8 of Exercise 98
   on page 241, using the Gaussian elimination
   method.


4. Repeat problems 3, 4, 8 and 9 of Exercise 99
   on page 244, using the Gaussian elimination
   method.