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Byju's Answer
Standard X
Mathematics
Solving Simultaneous Linear Equation Using Cramer's Rule
For what valu...
Question
For what values of p and q, the system of equations
2
x
+
p
y
+
6
z
=
8
x
+
2
y
+
q
z
=
5
x
+
y
+
3
z
=
4
has infinitely many solutions.
Enter the value of p
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Solution
Given equations are
2
x
+
p
y
+
6
z
=
8
x
+
2
y
+
q
z
=
5
x
+
y
+
3
z
=
4
Here
D
=
∣
∣ ∣
∣
2
p
6
1
2
q
1
1
3
∣
∣ ∣
∣
Applying
R
1
→
R
1
−
2
R
3
=
∣
∣ ∣
∣
0
p
−
2
0
1
2
q
1
1
3
∣
∣ ∣
∣
=
−
(
p
−
2
)
∣
∣
∣
1
q
1
3
∣
∣
∣
=
−
(
p
−
2
)
(
3
−
q
)
=
(
p
−
2
)
(
q
−
3
)
∴
D
1
=
∣
∣ ∣
∣
8
p
6
5
2
q
4
1
3
∣
∣ ∣
∣
Applying
C
1
→
C
1
−
4
C
2
&
C
3
→
C
3
−
3
C
2
D
1
=
∣
∣ ∣
∣
8
−
4
p
p
6
−
3
p
−
3
2
q
−
6
0
1
0
∣
∣ ∣
∣
Expanding along
R
3
, then
D
1
=
(
−
1
)
∣
∣
∣
8
−
4
p
6
−
3
p
−
3
q
−
6
∣
∣
∣
=
(
−
1
)
{
(
8
−
4
p
)
(
q
−
6
)
+
3
(
6
−
3
p
)
}
=
4
(
p
−
2
)
(
q
−
6
)
+
3
(
p
−
2
)
3
=
(
p
−
2
)
(
4
q
−
15
)
and
D
2
=
∣
∣ ∣
∣
2
8
6
1
5
q
1
4
3
∣
∣ ∣
∣
Applying
R
1
→
−
2
R
3
=
∣
∣ ∣
∣
0
0
0
1
5
q
1
4
3
∣
∣ ∣
∣
=
0
and
D
3
=
∣
∣ ∣
∣
2
p
8
1
2
5
1
1
4
∣
∣ ∣
∣
Applying
R
1
→
R
1
−
2
R
3
∴
D
3
=
∣
∣ ∣
∣
0
p
−
2
0
1
2
5
1
1
4
∣
∣ ∣
∣
Expanding along
R
1
then
D
3
=
−
(
p
−
2
)
∣
∣
∣
1
5
1
4
∣
∣
∣
C
3
=
(
p
−
2
)
By Cramer's rule :
x
=
D
1
D
,
y
=
D
2
D
,
z
=
D
3
D
For infinitely many solutions :
D
=
D
1
=
D
2
=
D
3
=
0
,
⇒
p
=
2
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0
Similar questions
Q.
The system of equations
2
x
+
p
y
+
6
z
=
8
x
+
2
y
+
q
z
=
5
x
+
y
+
3
z
=
4
has infinitely many solutions, then
p
=
?
Q.
For what value of
p
the system of equations
2
x
+
p
y
+
6
z
=
8
,
x
+
2
y
+
q
z
=
5
,
x
+
y
+
3
z
=
4
has unique solution.
Q.
For what values of
p
and
q
, the system of equations
2
x
+
p
y
+
6
z
=
8
,
x
+
2
y
+
q
z
=
5
,
x
+
y
+
3
z
=
4
has no solution. Give the value of q.
Q.
If the system of linear equations
2
x
+
p
y
+
6
z
=
8
x
+
2
y
+
q
z
=
5
x
+
y
+
3
z
=
4
has infinitely many solutions, then a possible pair of
p
and
q
is
Q.
If the system of equations
2
x
+
p
y
+
6
z
=
8
x
+
2
y
+
q
z
=
5
x
+
y
+
3
z
=
4
has no solution, then the value of
q
is?
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