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Question

(i) For which values of a and b does the following pair of linear equations have an infinite number of solutions?
2x+3y=7 ; (ab)x+(a+b)y=3a+b2
(ii) For which value of k will the following pair of linear equations have no solution?
3x+y=1 ; (2k1)x+(k1)y=2k+1

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Solution

i) For the pair of linear equations to have infinite number of solutions, a1a2=b1b2=c1c2

Given equation is 2x+3y=7
Thus a1=2,b1=3,c1=7
We know a2=ab,b2=a+b,c2=3ab+2

Therefore, 2ab=3a+b=7(3a+b2)
6a2b+4=7a+7b
a9b=4 ...(1)
and 9a+3b+6=7a7b

2a+4b=6 ...(2)

Solving (1) and (2), we get
a=5,b=1

ii) For the pair of linear equations to have infinite number of solutions,

a1a2=b1b2=c1c2

Given equation is 3x+y=1
Thus a1=3,b1=1,c1=1
We know a2=2k1,b2=k1,c2=2k+1

Therefore, 32k1=1(k1)1(2k+1)
3k3=2k1,2k+1k1k=2,k2

whenk=2 then equations will have no solution

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