Basic Engineering Mathematics, Fifth Edition

100 Basic Engineering Mathematics

Multiplying equation (1) by 4 gives

4 x+ 4 y+ 4 z= 16 (4)

Equation (2) – equation (4) gives

− 2 x− 7 y= 17 (5)

Similarly, multiplying equation (3) by 2 and then

adding this new equation to equation (2) will produce

another equation with onlyxandyinvolved.

Multiplying equation (3) by 2 gives

6 x− 4 y− 4 z=4(6)

Equation (2)+equation (6) gives

8 x− 7 y= 37 (7)

Rewriting equation (5) gives

− 2 x− 7 y= 17 (5)

Now we can use the previous method for solving

simultaneous equations in two unknowns.

Equation (7) – equation (5) gives 10 x= 20

from which, x= 2

(Note that 8x−− 2 x= 8 x+ 2 x= 10 x)

Substitutingx=2 into equation (5) gives

− 4 − 7 y= 17

from which, − 7 y= 17 + 4 = 21

and y=− 3

Substitutingx=2andy=−3 into equation (1) gives

2 − 3 +z= 4

from which, z= 5

Hence, the solution of the simultaneous equations is

x= 2 ,y=− 3 andz= 5.

Now try the following Practice Exercise

PracticeExercise 53 Simultaneous

equations in three unknowns (answers on

page 345)

In problems 1 to 9, solve the simultaneous equa-

tions in 3 unknowns.

1. x+ 2 y+ 4 z=16 2. 2x+y−z= 0
   2 x−y+ 5 z= 18 3 x+ 2 y+z= 4
   3 x+ 2 y+ 2 z= 14 5 x+ 3 y+ 2 z= 8


2. 3x+ 5 y+ 2 z=64.2x+ 4 y+ 5 z= 23
   x−y+ 3 z= 03 x−y− 2 z= 6
   2 + 7 y+ 3 z=− 34 x+ 2 y+ 5 z= 31


3. 2x+ 3 y+ 4 z=36 6. 4x+y+ 3 z= 31
   3 x+ 2 y+ 3 z= 29 2 x−y+ 2 z= 10
   x+y+z= 11 3 x+ 3 y− 2 z= 7


4. 5x+ 5 y− 4 z=37 8. 6x+ 7 y+ 8 z= 13
   2 x− 2 y+ 9 z= 20 3 x+y−z=− 11
   − 4 x+y+z=−14 2x− 2 y− 2 z=− 18


5. 3x+ 2 y+z= 14
   7 x+ 3 y+z= 22. 5
   4 x− 4 y−z=− 8. 5


6. Kirchhoff’s laws are used to determine the
   current equations in an electrical networkand
   result in the following:

i 1 + 8 i 2 + 3 i 3 =− 31

3 i 1 − 2 i 2 +i 3 =− 5

2 i 1 − 3 i 2 + 2 i 3 = 6

Determine the values ofi 1 ,i 2 andi 3

11. The forces in three members of a frame-
    work areF 1 ,F 2 andF 3. They are related by
    following simultaneous equations.

1. 4 F 1 + 2. 8 F 2 + 2. 8 F 3 = 5. 6

4. 2 F 1 − 1. 4 F 2 + 5. 6 F 3 = 35. 0

4. 2 F 1 + 2. 8 F 2 − 1. 4 F 3 =− 5. 6

Find the values ofF 1 ,F 2 andF 3