untitled

Now, if there is no constant terms in both the equations of a system of equations ;

i.e.,c 1 c 2 0 , if with reference to the above discussion from (i), if
2

1

2

1

b

b

a

a

z the

system of equations are always consistent and independent of each other. In that
case, there will be only one (unique) solution.

From (ii) and (iii) if
2

1

2

1

b

b

a

a

, the system of equations are consistent and

dependent of each other. In that case, there will be infinite number of solutions.
Example : Explain whether the following sy stem of equations are consistent /
inconsistent, dependent/ independent of each other and indicate the number of
solutions in each case.
(a)x 3 y 1 (b) 2 x 5 y 3 (c) 3 x 5 y 7
2 x 6 y 2 x 3 y 1 6 x 10 y 15
Solution :

(a) Given system of equations are :
¿

¾

½





2 6 2

3 1

x y

x y

Ratio of the coefficients of x is

2

1

Ratio of the coefficients of y is

6

3

or

2

1

Ratio of constant terms is

2

1

2

1

6

3

2

1

?

Therefore, the system of equations are consistent and mutually dependent. The
system of equations have infinite number of solutions.

(b) Given system of equations are :
°¿

°

¾

½





3 1

2 5 3

x y

x y

Ratio of the coefficients of x is

1

2

Ratio of the coefficients of y is

3

 5

? we have,
3

5

1

2 

z

Therefore, the system of equations are consistent and mutually independent. The
system of equations have only one (unique) solution.