3. On comparing the ratios find out whether the following pair of linear
equations are consistent, or inconsistent.
3/2x+5/3y=7 ;9x-10y=4
Answers
Answer: ∵
a
2
a
1
=
b
2
b
1
=
c
2
c
1
,
Step-by-step explanation:
(i)
a
2
a
1
=
2
3
,
b
2
b
1
=
−3
2
,
c
2
c
1
=
7
5
∵
a
2
a
1
=
b
2
b
1
=
c
2
c
1
,
the lines intersect and have an unique consistent solution.
(ii)
a
2
a
1
=
4
2
=
2
1
,
b
2
b
1
=
−6
−3
=
2
1
,
c
2
c
1
=
9
8
∵
a
2
a
1
=
b
2
b
1
=
c
2
c
1
,
the lines are parallel and have no solutions, i.e. the equations are an inconsistent pair
(iii)
a
2
a
1
=
9
2
3
=
6
1
,
b
2
b
1
=
−10
3
5
=−
6
1
,
c
2
c
1
=
14
7
=
2
1
∵
a
2
a
1
=
b
2
b
1
=
c
2
c
1
,
the lines intersect and have an unique consistent solution.
(iv)
a
2
a
1
=
−10
5
=−
2
1
,
b
2
b
1
=
6
−3
=−
2
1
,
c
2
c
1
=
−22
11
=−
2
1
∵
a
2
a
1
=
b
2
b
1
=
c
2
c
1
,
the lines are coincident and have infinitely many solutions. The equations form a consistent pair of equations.
(v)
a
2
a
1
=
2
3
4
=
3
2
,
b
2
b
1
=
3
2
,
c
2
c
1
=
12
8
=
3
2
∵
a
2
a
1
=
b
2
b
1
=
c
2
c
1
,
the lines are coincident and have infinitely many solutions. The equations form a consistent pair of equations.(i)
a
2
a
1
=
2
3
,
b
2
b
1
=
−3
2
,
c
2
c
1
=
7
5
∵
a
2
a
1
=
b
2
b
1
=
c
2
c
1
,
the lines intersect and have an unique consistent solution.
(ii)
a
2
a
1
=
4
2
=
2
1
,
b
2
b
1
=
−6
−3
=
2
1
,
c
2
c
1
=
9
8
∵
a
2
a
1
=
b
2
b
1
=
c
2
c
1
,
the lines are parallel and have no solutions, i.e. the equations are an inconsistent pair
(iii)
a
2
a
1
=
9
2
3
=
6
1
,
b
2
b
1
=
−10
3
5
=−
6
1
,
c
2
c
1
=
14
7
=
2
1
∵
a
2
a
1
=
b
2
b
1
=
c
2
c
1
,
the lines intersect and have an unique consistent solution.
(iv)
a
2
a
1
=
−10
5
=−
2
1
,
b
2
b
1
=
6
−3
=−
2
1
,
c
2
c
1
=
−22
11
=−
2
1
∵
a
2
a
1
=
b
2
b
1
=
c
2
c
1
,
the lines are coincident and have infinitely many solutions. The equations form a consistent pair of equations.
(v)
a
2
a
1
=
2
3
4
=
3
2
,
b
2
b
1
=
3
2
,
c
2
c
1
=
12
8
=
3
2
∵
a
2
a
1
=
b
2
b
1
=
c
2
c
1
,
the lines are coincident and have infinitely many solutions. The equations form a consistent pair of equations.