find the value of K for which of the following system of linear equation has infinite no. of solutions ?
x+( k+1) = 5
(k+1)x + 9y=(8k-1)
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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
=1,b
1
=k+1,c
1
=5
and a
2
=k+1,b
2
=9 and c
2
=8k−1
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 solutions, if
k+1
1
=
9
k+1
=
8k−1
5
⇒
k+1
1
=
9
k+1
and
9
k+1
=
8k−1
5
⇒(k+1)
2
=9 and (k+1)(8k−1)=45
Now, (k+1)
2
=9
⇒k+1=±3⇒k=2,−4
We observe that k=2 satisfies the equation (k+1)(8k−1)=45 but, k=−4 does not satisfy it.
So, for k=2 the given system of the linear equations has infinite solutions.
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