For what value of the following pair of linear equation has infinitely many solutions?

`a=3, b=1``a=1, b=3``a=1, b=1``a=3 , b=3`

Answer : A

Solution : Given pair of linear equations are
x + 2y = 1 `" " ` ...(i)
and `(a - b ) x + (a + b) y = a + b - 2 " " ...(ii)`
On comparing with `ax+by+c=0`, we get
`a_(1)=1, b_(1)=2` and `c_(1)=-1" "` [ from Eq. (i)]
`a_(2)=(a-b),b_(2)=(a+b) " " `[from Eq. (ii)]
and `" " c_(2) =-(a+b-2)`
For infinitely many solutions of the pairs of linear equations,
`(a_(1))/(a_(2))=(b_(1))/(b_(2))=(c_(1))/(c_(2))`
`rArr " " (1)/(a-b) =(2)/(a+b)=(-1)/(-(a+b-2))`
Taking first two parts,
`(1)/(a-b)=(2)/(a+b)`
`rArr " " a+b = 2a-2b`
`rArr " " 2a-a=2b+b`
`rArr " " a = 3b " " ...(iii)`
Taking last two parts,
`(2)/(a+b)=(1)/(a+b-2)`
`rArr" " 2a+2b-4=a+b`
`rArr " " a+b=4 " " ...(iv)`
Now, put the value of a from Eq. (iii) in Eq. (iv), we get
`3b+b=4`
`rArr" "4b=4`
`rArr " " b = 1`
Put the value of b in Eq. (iii), we get
`a=3xx1`
`rArr " " a=3`
So, the values (a,b)=(3,1) satisfies all the parts. Hence, required values of a and b are 3 and 1 respectively for which the given pair of linear equations has infinitely many solutions.

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Updated On: 27-06-2022

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