Solving systems of simultaneous linear equations is another task performed by engineers. Consider an example of a system of simultaneous linear equations in 2 variables:
EXAMPLE:
| 3x 2x |
– – |
y 4y |
= = |
7 – 2 |
|||||
|---|---|---|---|---|---|---|---|---|---|
| ( |
3x 2x |
– – |
y 4y |
= = |
7 – 2 |
) |
– 3/2 |
(1) (2) |
|
| – |
3x 3x |
– + |
y 6y |
= = |
7 3 |
(3) |
|||
| 5y y |
= = |
10 2 |
(4) |
||||||
| Substituting (4) in (1), we get | |||||||||
| 3x | – | 2 3x x |
= = = |
7 9 3 |
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In general, a system of simultaneous linear equations in n variables
| a11x1 a21x1 . . . an1x1 |
+ + + |
a12x2 a22x2 . . . an2x2 |
+ . . . + + . . . + + . . . + |
a1nxn a2nxn . . . annxn |
= = = |
b1 b2 . . . bn |
|---|
|
Sa1j xj = b1 . . . Sanj xj= bn |
ü ÷ ý ÷ þ |
Saijxj = bi, i = 1, 2, . . . , n; j = 1, 2, . . . , n |
|---|
A system of simultaneous linear equations in n variables may also be represented in matrix form as follows:
An x n x Xn x 1 = Bn x 1
where
| An x n = |
æ ç ç ç ç è |
a11 a21 . . . an1 |
a12 a22 . . . an2 |
. . . . . . . . . |
a1n a2n . . . ann |
ö ÷ ÷ ÷ ÷ ø |
|---|
| Xn x 1 = |
æ ç ç ç ç è |
x1 x2 . . . xn |
ö ÷ ÷ ÷ ÷ ø |
and | Bn x 1 = |
æ ç ç ç ç è |
b1 b2 . . . bn |
ö ÷ ÷ ÷ ÷ ø |
|---|
We shall look at three techniques for solving systems of simultaneous linear equations, namely , Gauss–Jordan Method and the Cramer's Rule.