The Chemistry Maths Book, Second Edition

17.4 The solution of linear equations 485

The sum of (2) and (3) is now equal to twice (1), but this means that equation (1)

contains no information not already contained in the other two equations. The

equations are said to be linearly dependent, and each equation can be expressed as a

linear combination of the others. We have effectively only two equations in three

unknowns. For example, solving (1) and (2), or any pair, for xand yin terms of zgives

(17.33)

and this is a solution of the system (17.32) for every value of z.

0 Exercise 17

Homogeneous equations

When at least one of the quantitiesb

k

on the right sides of equations (17.26) is not

zero, the equations are called inhomogeneous equations. When all theb

k

are zero,

they are calledhomogeneous equations:

a

11

x

1

1 + 1 a

12

x

2

1 + 1 a

13

x

3

1 + 1  1 + 1 a

1 n

x

n

1 = 10

a

21

x

1

1 + 1 a

22

x

2

1 + 1 a

23

x

3

1 + 1  1 + 1 a

2 n

x

n

1 = 10

a

31

x

1

1 + 1 a

32

x

2

1 + 1 a

33

x

3

1 + 1  1 +a

3 n

x

n

1 = 10 (17.34)

a

n 1

x

1

1 + 1 a

n 2

x

2

1 + 1 a

n 3

x

3

1 + 1  1 + 1 a

nn

x

n

1 = 10

Only the zero solution (17.30) exists ifD 1 ≠ 10 , but other solutions also exist when

D 1 = 10. For example, the system

(1) 2x 1 + 12 y 1 + z 1 = 10

(2) x+ 12 y 1 − 12 z 1 = 10 (17.35)

(3) 3x 1 + 12 y 1 + 14 z 1 = 10

hasD 1 = 10 and, like (17.32), the equations are linearly dependent. One solution is the

trivial (zero) solutionx 1 = 1 y 1 = 1 z 1 = 10. Nonzerosolutions are obtained by solving any

pair of the equations for xand yin terms of z:

(17.36)

for all values of z. A uniquesolution is obtained only if a further, independent, relation

amongst x, y, and zis known.

This example demonstrates one of the most important theorems of systems of

linear equations:

A system of homogeneous linear equations has nontrivial

solution only if the determinant of the coefficients is zero.

xz y

z

=− , 3 =

5

2

xzyz=−,13 3 = −

1

2

() 516