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12 21 12 21

1

bc bc ab ab

x





, or

12 21

12 21

ab ab

bc bc

x





Again,
12 21 12 21

1

ca ca ab ab

y





, or

12 21

12 2 1

ab ab

ca ca

y





? The solution of the given equations : ̧ ̧
¹

·

̈ ̈

©

§









12 21

1 2 21

12 21

12 21

ab ab

ca ca

ab ab

bc bc

(x,y) ,

We observe :

Equations Relation between xand y Illustration

0

0

2 2 2

1 1 1

 

 

ax by c

ax by c

12 21 12 21 12 21

1

ca ca ab ab

y

bc bc

x







x y 1

2 2 2 2 2

1 1 1 1 1

a b c a b

a b c a b

[N.B. : The method of cross-multiplication can also be applied by keeping the
constant terms of both equations on the right hand side. In that case, changes of sign
will occur ; but the solution will remain same.]

Activity : If the system of equations

¿

¾

½



 

3 0

4 7 0

x y

x y

are expressed as the system of equations

¿

¾

½

 

 

0

0

2 2 2

1 1 1

ax by c

ax by c

,

find the values of a 1 ,b 1 ,c 1 ,a 2 ,b 2 ,c 2.

Example 3. Solve by the method of cross-multiplication : 6 xy 1
3 x 2 y 13
Solution : Making the right hand side of the equations 0 (zero) by transposition, we get,

3 2 13 0

6 1 0

 

 

x y

x y comparing the equations with

¿

¾

½

 

 

0

0

2 2 2

1 1 2 1

ax by c

ax by c

respectively,

we get, a 1 6 ,b 1  1 ,c 1  1

a 2 3 ,b 2 2 ,c 2  13

By the method of cross-multiplication, we get,

12 21 12 21 12 21

1

ca ca ab ab

y

bc bc

x







Illustration :

x y 1

2 2 2 2 2

1 1 1 1 1

a b c a b

a b c a b