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 (a b)x+(a+b)y=3a+b 2`(ii) For which value of k will the following pair of linear equations have

Answer 1

The given equations are
` 2x + 3y - 7 = 0," " `… (i)
` ( a - b ) x + ( a + b ) y + ( 2 - 3a - b ) = 0 " " ` … (ii)
These equations are of the form
` a_ 1 x + b_ 1y + c _ 1 = 0 and a_ 2 x + b _ 2 y + c_ 2 = 0 `,
where ` a_ 1 = 2, b_ 1 = 3, c_ 1 = - 7`
and ` a_ 2 = (a- b ), b_ 2 = (a + b ), c_ 2 = ( 2 - 3a - b )`
` therefore (a_ 1 )/(a_ 2) = (2)/((a- b)), (b_ 1)/(b_ 2 ) = (3)/((a+ b ) ) and (c_ 1 ) /(c_ 2) = (-7)/(( 2 - 3a - b )) = (7)/((3a + b - 2 ))`
Let the given system of equations have infinitely many solutions.
Then , ` (a_ 1 )/(a _ 2 ) = (b_ 1 )/( b_ 2 ) = (c _ 1 )/( c_ 2 )`
` rArr ( 2 ) /(( a- b )) = ( 3) /(( a + b )) = ( 7)/(( 3a + b - 2 ))`
` rArr ( 2) /(( a- b ) ) = ( 3) /(( a+ b )) and ( 3) /(( a+ b )) = ( 7) /(( 3a + b - 2 ))`
` rArr 2 a + 2 b = 3 a - 3 b and 9a + 3 b - 6 = 7a + 7b `
` rArr a - 5b = 0 and 2a - 4 b = 6 `
` rArr a - 5 b = 0 and a- 2b = 3`.
On solving ` a - 5 b = 0 and a - 2 b = 3,` we get ` a =5 and b = 1 `
Hence, the required values are ` a = 5 and b = 1 `.