For what solutions of p and q, the following pair of linear equations will have infinitely many solutions
Answers
Answer:
This system of equation is of the form
a
1
x+b
1
y=c
1
a
2
x+b
2
y=c
2
where a
1
=2,b
1
=3,c
1
=7
and a
2
=p+q,b
2
=2p−q and c
2
=21
For infinitely many solutions, we must have
a
2
a
1
=
b
2
b
1
=
c
2
c
1
The given system of equations will have infinite number of solutions, if
p+q
2
=
2p−q
3
=
21
7
⇒
p+q
2
=
2p−q
3
=
3
1
⇒
p+q
2
=
3
1
and
2p−q
3
=
3
1
⇒p+q=6 and 2p−q=9
⇒(p+q)+(2p−q)=6+9 [On adding]
⇒3p=15
⇒p=15
Putting p=5 in p+q=6 or 2p−q=9, we get q=1.
Hence, the givens system of equations will have infinitely many solutions, if p=5 and q=1.
hey dude this is ur answer
Given equations:-
2x+3y=9
(p+q)x+(2p−q)y=3(p+q+1)
Here,
a
2
a
1
=
p+q
2
,
b
2
b
1
=
2p−q
3
,
c
2
c
1
=
3(p+q+1)
9
For a pair of linear equations to have infinitely many solutions:
a
2
a
1
=
b
2
b
1
=
c
2
c
1
So, we need,
p+q
2
=
2p−q
3
=
3(p+q+1)
9
or,
p+q
2
=
2p−q
3
=>2(2p−q)=3(p+q)
=>4p−2q=3p+3q
=>p=5q....(i)
Also,
2p−q
3
=
3(p+q+1)
9
=>9(p+q+1)=9(2p−q)
=>p+q+1=2p−q
=>2p−p=q+q+1
=>p=2q+1
Substituting(i),wehave,
5q=2q+1
=>q=
3
1
Also,p=5q=5(
3
1
)=
3
5
∴p=
3
5
andq=
3
1