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:
(abcd)(xy)=(ef),
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:
(−6132)(xy)=(19−22)
2. Matrix equations in the form
(abcd)(xy)=(ef)
can be solved for the unknowns x and y as follows.
(a) Let A=(abcd) and find A-1.
(b) Multiply both sides of the equation by A-1.
A−1(abcd)(xy)=A−1(ef)
(c) A−1A(xy)=A−1(ef) I(xy)=A−1(ef) ↑A−1A=I=(1001) (xy)=A−1(ef) (xy)=1ad−bc(d−b−ca)(ef)
(c) A−1A(xy)=A−1(ef) I(xy)=A−1(ef) ↑A−1A=I=(1001) (xy)=A−1(ef) (xy)=1ad−bc(d−b−ca)(ef)
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
(2375)(xy)=(51)←write the simultaneousequations in matrix form.LetA=(2375)A−1=1ad−bc(d−b−ca)A−1=110−21(5−3−72)A−1=1−11(5−3−72)(xy)=1−11(5−3−72)(51)←(xy)=A−1(ef)(xy)=1−11(5×5+(−3)×1−7×5+2×1)(xy)=1−11(22−33)(xy)=(−2211−33−11)=(−23)∴