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 :
Sol :
2x + y = 5 , 3x + 2y = 8
Comparing equation
(i)
with
a1x
+
b1y
+
c1
=
0
and equation (ii) with
a2x
+
b2y
+
c2
=
0
We get
a1
=
2
a2
=
3
b1
=
1
b2
=
2
c1
=
–5
c2
=
–8
a1
a2
2
3
=
... (iii)
b1
b2
=
... (iv)
c1
c2
-5
-8
=
... (v)
a1
a2
≠
b1
b2
From (iii), (iv) and (v)
2x
+
y
–
5
=
0
3x
+
2y
–
8
=
0
... (i)
... (ii)
1
2
5
8
=
x
b1
c2
–
b2
c1
=
y
c1
a2
–
c2
a1
=
1
a1
b2
–
a2
b1
∴
x
–8
–
(–
10)
=
y
–15
–(
–
16)
=
1
4
–
3
∴
x
2
=
1
⇒
x
=
2
∴
y
1
=
1
⇒
y
=
1
Solution is x = 2 and y = 1.
Equations has unique solution (Consistent)
Equations has
no solution (Inconsistent)
Equations has infinite solutions
(Consistent)
∴
x
2
=
y
1
=
1
1