Higher Engineering Mathematics, Sixth Edition

The solution of simultaneous equations by matrices and determinants 243

⎛

⎝

100

010

001

⎞

⎠×

⎛

⎝

x

y

z

⎞

⎠=

1

35

×

⎛

⎝

( 14 × 4 )+( 0 × 33 )+( 7 × 2 )

( 16 × 4 )+((− 5 )× 33 )+((− 2 )× 2 )

( 5 × 4 )+( 5 × 33 )+((− 5 )× 2 )

⎞

⎠

⎛

⎝

x

y

z

⎞

⎠=^1

35

⎛

⎝

( 14 × 4 )+( 0 × 33 )+( 7 × 2 )

( 16 × 4 )+((− 5 )× 33 )+((− 2 )× 2 )

( 5 × 4 )+( 5 × 33 )+((− 5 )× 2 )

⎞

⎠

=

1

35

⎛

⎝

70

− 105

175

⎞

⎠

=

⎛

⎝

2

− 3

5

⎞

⎠

(v) By comparing corresponding elements, x= 2 ,

y=− 3 , z= 5 , which can be checked in the

original equations.

Now try the following exercise

Exercise 98 Further problems on solving

simultaneous equations using matrices

In Problems 1 to 5 usematrices to solve the

simultaneous equations given.

1. 3x+ 4 y= 0
   2 x+ 5 y+ 7 =0[x= 4 ,y=−3]


2. 2p+ 5 q+ 14. 6 = 0
   3. 1 p+ 1. 7 q+ 2. 06 = 0
   [p= 1. 2 ,q=− 3 .4]


3. x+ 2 y+ 3 z= 5
   2 x− 3 y−z= 3
   − 3 x+ 4 y+ 5 z= 3
   [x= 1 ,y=− 1 ,z=2]


4. 3 a+ 4 b− 3 c= 2
   − 2 a+ 2 b+ 2 c= 15
   7 a− 5 b+ 4 c= 26
   [a= 2. 5 ,b= 3. 5 ,c= 6 .5]


5. p+ 2 q+ 3 r+ 7. 8 = 0
   2 p+ 5 q−r− 1. 4 = 0
   5 p−q+ 7 r− 3. 5 = 0
   [p= 4. 1 ,q=− 1. 9 ,r=− 2 .7]
   6. In two closed loops of an electrical circuit, the
   currents flowingare givenby the simultaneous
   equations:
   I 1 + 2 I 2 + 4 = 0
   5 I 1 + 3 I 2 − 1 = 0
   Use matrices to solve forI 1 andI 2.
   [I 1 =2,I 2 =−3]
   7. The relationship between the displacement,s,
   velocity,v, and acceleration,a, of a piston is
   given by the equations:
   s+ 2 v+ 2 a= 4
   3 s−v+ 4 a= 25
   3 s+ 2 v−a=− 4
   Use matrices to determine the values ofs,v
   anda.
   [s= 2 ,v=− 3 ,a=4]


6. In a mechanical system, acceleration x ̈,
   velocityx ̇and distancexare related by the
   simultaneous equations:


7. 4 x ̈+ 7. 0 x ̇− 13. 2 x=− 11. 39
   − 6. 0 x ̈+ 4. 0 x ̇+ 3. 5 x= 4. 98


8. 7 x ̈+ 6. 0 x ̇+ 7. 1 x= 15. 91
   Use matrices to find the values ofx ̈,x ̇andx.
   [x ̈= 0 .5,x ̇= 0 .77,x= 1 .4]

23.2 Solution of simultaneous

equations by determinants

(a) When solving linear simultaneous equations in

two unknowns using determinants:

(i) write the equations in the form

a 1 x+b 1 y+c 1 = 0

a 2 x+b 2 y+c 2 = 0

and then

(ii) the solution is given by

x

Dx

=

−y

Dy

=

1

D

where Dx=

∣

∣

∣

∣

∣

b 1 c 1

b 2 c 2

∣

∣

∣

∣

∣

i.e. the determinant of the coefficients left

when thex-column is covered up,