100 Basic Engineering Mathematics
Multiplying equation (1) by 4 gives
4x + 4y + 4z = 16
(4)
Equation (2) – equation (4) gives
−2x − 7y = 17
(5)
Similarly, multiplying equation (3) by 2 and then
adding this new equation to equation (2) will produce
another equation with only x and y involved.
Multiplying equation (3) by 2 gives
6x − 4y − 4z = 4
( 6 )
Equation (2) + equation (6) gives
8x − 7y = 37
(7)
Rewriting equation (5) gives
−2x − 7y = 17
(5)
Now we can use the previous method for solving
simultaneous equations in two unknowns.
Equation (7) – equation (5) gives
10x = 20
from which,
x = 2
(Note that 8x − −2x = 8x + 2x = 10x)
Substituting x = 2 into equation (5) gives
−4 − 7y = 17
from which,
−7y = 17 + 4 = 21
and
y = −3
Substituting x = 2 and y = −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 and z = 5.
Now try the following Practice Exercise
Practice Exercise 53 Simultaneous
equations in three unknowns (answers on
page 345)
In problems 1 to 9, solve the simultaneous equations in 3 unknowns.
1. x + 2y + 4z = 16 2. 2x + y − z = 0
2x − y + 5z = 18
3x + 2y + z = 4
3x + 2y + 2z = 14
5x + 3y + 2z = 8
3. 3x + 5y + 2z = 6 4. 2x + 4y + 5z = 23
x − y + 3z = 0
3 x − y − 2z = 6
2 + 7y + 3z = −3
4x + 2y + 5z = 31
5. 2x + 3y + 4z = 36 6. 4x + y + 3z = 31
3x + 2y + 3z = 29
2x − y + 2z = 10
x + y + z = 11
3x + 3y − 2z = 7
7. 5x + 5y − 4z = 37 8. 6x + 7y + 8z = 13
2x − 2y + 9z = 20
3x + y − z = −11
−4x + y + z = −14
2x − 2y − 2z = −18
9. 3x + 2y + z = 14
7x + 3y + z = 22.5
4x − 4y − z = −8.5
10. Kirchhoff’s laws are used to determine the
current equations in an electrical network and
result in the following:
i 1 + 8i 2 + 3i 3 = −31
3i 1 − 2i 2 + i 3 = −5
2i 1 − 3i 2 + 2i 3 = 6
Determine the values of i 1 , i 2 and i 3
11. The forces in three members of a framework are F 1 , F 2 and F 3 . They are related by
following simultaneous equations.
1.4F 1 + 2.8F 2 + 2.8F 3 = 5.6
4.2F 1 − 1.4F 2 + 5.6F 3 = 35.0
4.2F 1 + 2.8F 2 − 1.4F 3 = −5.6
Find the values of F 1 , F 2 and F 3
Multiplying equation (1) by 4 gives
4x + 4y + 4z = 16
(4)
Equation (2) – equation (4) gives
−2x − 7y = 17
(5)
Similarly, multiplying equation (3) by 2 and then
adding this new equation to equation (2) will produce
another equation with only x and y involved.
Multiplying equation (3) by 2 gives
6x − 4y − 4z = 4
( 6 )
Equation (2) + equation (6) gives
8x − 7y = 37
(7)
Rewriting equation (5) gives
−2x − 7y = 17
(5)
Now we can use the previous method for solving
simultaneous equations in two unknowns.
Equation (7) – equation (5) gives
10x = 20
from which,
x = 2
(Note that 8x − −2x = 8x + 2x = 10x)
Substituting x = 2 into equation (5) gives
−4 − 7y = 17
from which,
−7y = 17 + 4 = 21
and
y = −3
Substituting x = 2 and y = −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 and z = 5.
Now try the following Practice Exercise
Practice Exercise 53 Simultaneous
equations in three unknowns (answers on
page 345)
In problems 1 to 9, solve the simultaneous equations in 3 unknowns.
1. x + 2y + 4z = 16 2. 2x + y − z = 0
2x − y + 5z = 18
3x + 2y + z = 4
3x + 2y + 2z = 14
5x + 3y + 2z = 8
3. 3x + 5y + 2z = 6 4. 2x + 4y + 5z = 23
x − y + 3z = 0
3 x − y − 2z = 6
2 + 7y + 3z = −3
4x + 2y + 5z = 31
5. 2x + 3y + 4z = 36 6. 4x + y + 3z = 31
3x + 2y + 3z = 29
2x − y + 2z = 10
x + y + z = 11
3x + 3y − 2z = 7
7. 5x + 5y − 4z = 37 8. 6x + 7y + 8z = 13
2x − 2y + 9z = 20
3x + y − z = −11
−4x + y + z = −14
2x − 2y − 2z = −18
9. 3x + 2y + z = 14
7x + 3y + z = 22.5
4x − 4y − z = −8.5
10. Kirchhoff’s laws are used to determine the
current equations in an electrical network and
result in the following:
i 1 + 8i 2 + 3i 3 = −31
3i 1 − 2i 2 + i 3 = −5
2i 1 − 3i 2 + 2i 3 = 6
Determine the values of i 1 , i 2 and i 3
11. The forces in three members of a framework are F 1 , F 2 and F 3 . They are related by
following simultaneous equations.
1.4F 1 + 2.8F 2 + 2.8F 3 = 5.6
4.2F 1 − 1.4F 2 + 5.6F 3 = 35.0
4.2F 1 + 2.8F 2 − 1.4F 3 = −5.6
Find the values of F 1 , F 2 and F 3