Chapter 23

The solution of simultaneous

equationsbymatricesand

determinants

23.1 Solution of simultaneous

equations by matrices

(a) The procedure for solving linear simultaneous
equations intwo unknowns using matricesis:
(i) write the equations in the form
a 1 x+b 1 y=c 1
a 2 x+b 2 y=c 2
(ii) write the matrix equation corresponding to
these equations,
i.e.

(

a 1 b 1

a 2 b 2

)

×

(

x

y

)

=

(

c 1

c 2

)

(iii) determine the inverse matrix of

(

a 1 b 1

a 2 b 2

)

i.e.

1

a 1 b 2 −b 1 a 2

(

b 2 −b 1

−a 2 a 1

)

(from Chapter 22)

(iv) multiplyeach side of (ii) by the inverse

matrix, and

(v) solve forxandyby equating corresponding

elements.

Problem 1. Use matrices to solve the

simultaneous equations:

3 x+ 5 y− 7 =0(1)

4 x− 3 y− 19 =0(2)

(i) Writing the equations in thea 1 x+b 1 y=cform

gives:

3 x+ 5 y= 7

4 x− 3 y= 19

(ii) The matrix equation is

(

35

4 − 3

)

×

(

x

y

)

=

(

7

19

)

(iii) The inverse of matrix

(

35

4 − 3

)

is

1

3 ×(− 3 )− 5 × 4

(

− 3 − 5

− 43

)

i.e.

⎛

⎜

⎝

3

29

5

29

4

29

− 3

29

⎞

⎟

⎠

(iv) Multiplying each side of (ii) by (iii) and

remembering thatA×A−^1 =I, the unit matrix,

gives:

(

10

01

)(

x

y

)

=

⎛

⎜

⎜

⎝

3

29

5

29

4

29

− 3

29

⎞

⎟

⎟

⎠×

(

7

19

)