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(i) For which values of a and b does the following pair of linear equations have an infinite number of solutions? 2x + 3y = 7 (a - b) x + (a + b) y = 3a + b - 2

Q2.    (i) For which values of a and b does the following pair of linear equations have an infinite number of solutions?
                \\2x + 3y = 7 \\(a - b) x + (a + b) y = 3a + b - 2

Answers (1)
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Given     equations,

\\2x + 3y = 7 \\(a - b) x + (a + b) y = 3a + b - 2

As we know, the condition for equations a_1x+b_1y+c_1=0\:and\:a_2x+b_2y+c_2=0  to have an infinite solution is

\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}

So, Comparing these equations with, a_1x+b_1y+c_1=0\:and\:a_2x+b_2y+c_2=0, we get

\frac{2}{a-b}=\frac{3}{a+b}=\frac{7}{3a+b-2}

From here we get,

\frac{2}{a-b}=\frac{3}{a+b}

\Rightarrow 2(a+b)=3(a-b)

\Rightarrow 2a+2b=3a-3b

\Rightarrow a-5b=0.........(1)

Also,

\frac{2}{a-b}=\frac{7}{3a+b-2}

\Rightarrow 2(3a+b-2)=7(a-b)

\Rightarrow 6a+2b-4=7a-7b

\Rightarrow a-9b+4=0...........(2)

Now, Subtracting (2) from (1) we get

\Rightarrow 4b-4=0

\Rightarrow b=1

Substituting this value in (1) 

\Rightarrow a-5(1)=0

\Rightarrow a=5

Hence, a=5\:and\:b=1.

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