plz answer this....
Answers
Answer:
Given the following equations:
x
+
2
y
=
3
∴
x
+
2
y
−
3
=
0
..........
1
5
x
+
k
y
+
7
=
0
..........
2
The following are a pair of standard linear equations:
a
1
x
+
b
1
y
+
c
1
=
0
..........
3
a
2
x
+
b
2
y
+
c
2
=
0
..........
4
Comparing equations
1
and
3
, we get:
a
1
=
1
,
b
1
=
2
,
c
1
=
−
3
Comparing equations
2
and
4
, we get:
a
2
=
5
,
b
2
=
k
,
c
2
=
7
i
We need to find the value of
k
such that the given system has a unique solution.
For equations
1
and
2
to have a unique solution, we must satisfy the following condition:
a
1
a
2
≠
b
1
b
2
Substituting the given values, we see that:
1
5
≠
2
k
∴
k
≠
10
Therefore, all values of
k
except
k
=
10
will result in the given system of equations having a unique solution.
i
i
We need to find the value of
k
such that the given system has no solutions.
For equations
1
and
2
to have no solutions, we must satisfy the following condition:
a
1
a
2
=
b
1
b
2
≠
c
1
c
2
Substituting the given values, we see that:
1
5
=
2
k
≠
−
3
7
∴
1
5
=
2
k
and
2
k
≠
−
3
7
∴
k
=
10
and
k
≠
−
14
3
.
Therefore,
k
=
10
will result in the given system of equations having no solutions at all.
Furthermore, we note that
5
7
≠
−
3
7
Hence, there are no values of
k
such that the given system of equations has an infinite number of solutions