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4.8 Solving Simultaneous Linear Equations using Matrices

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4.8 Solving Simultaneous Linear Equations using Matrices
1. Two simultaneous linear equations can be written in the matrix equation form.
For example, in the simultaneous equations:
ax + by = e
cx + dy = f
can be written in the matrix form as follows:

( a b c d ) ( x y ) = ( e f ) ,

Where a, b, c, d, e and f are constant while x and y are unknowns.

Example 1:
Write the following simultaneous linear equations in the matrix form.
y– 6x – 19 = 0
2y + 3x + 22 = 0

Solution:
– 6x + y = 19
3x + 2y = – 22
The matrix form is:
( − 6 1 3 2 ) ( x y ) = ( 19 − 22 )



2. Matrix equations in the form ( a b c d ) ( x y ) = ( e f )
can be solved for the unknowns x and y as follows.

(a) Let A = ( a b c d ) and find A-1.

(b) Multiply both sides of the equation by A-1.
A − 1 ( a b c d ) ( x y ) = A − 1 ( e f )

(c)  A − 1 A ( x y ) = A − 1 ( e f )   I ( x y ) = A − 1 ( e f )   ↑ A − 1 A = I = ( 1 0 0 1 )   ( x y ) = A − 1 ( e f )   ( x y ) = 1 a d − b c ( d − b − c a ) ( e f )



Example 2:
Solve the following simultaneous equations by using the matrix method.
2x = 5 – 3y
7x = 1 – 5y

Solution:
2x + 3y = 5
7x + 5y = 1 
( 2 3 7 5 ) ( x y ) = ( 5 1 ) ← write the simultaneous equations in matrix form . Let A = ( 2 3 7 5 ) A − 1 = 1 a d − b c ( d − b − c a ) A − 1 = 1 10 − 21 ( 5 − 3 − 7 2 ) A − 1 = 1 − 11 ( 5 − 3 − 7 2 ) ( x y ) = 1 − 11 ( 5 − 3 − 7 2 ) ( 5 1 ) ← ( x y ) = A − 1 ( e f ) ( x y ) = 1 − 11 ( 5 × 5 + ( − 3 ) × 1 − 7 × 5 + 2 × 1 ) ( x y ) = 1 − 11 ( 22 − 33 ) ( x y ) = ( − 22 11 − 33 − 11 ) = ( − 2 3 ) ∴ x = − 2 , y = 3.