Q. Which of the following pairs of linear equations has unique solution, no solution, or infinitely many solutions. In case there is a unique solution, find it by using cross multiplication method :
3x – 5y = 20 ; 6x – 10y = 40
Sol :
3
Comparing equation
(i)
with
a1x
+
b1y
+
c1
=
0
and equation (ii) with
a2x
+
b2y
+
c2
=
0
We get
a1
=
3
a2
=
6
b1
=
-5
b2
=
-10
c1
=
-20
c2
=
-40
a1
a2
3
6
=
... (iii)
b1
b2
-5
-10
=
... (iv)
=
c1
c2
- 20
- 40
=
... (v)
a1
a2
=
b1
b2
From (iii), (iv) and (v)
x
–
5 y
–
20
=
0
6x
–
10 y
–
40
=
0
... (i)
... (ii)
1
2
infinitely many solutions.
1
2
=
1
2
=
=
c1
c2
Equations has unique solution (Consistent)
Equations has
no solution (Inconsistent)
Equations has infinite solutions
(Consistent)