For the following system of equations determine the value of `k` for which the given system has infinitely many solutions:`k x+3y=k-3,\ \ \ \ 12 x+k y=k`

Explanation

The given system of equations is
` k x + 3y + ( 3 - k ) = 0 `,
` 12 x + k y - k = 0 `.
These equations are of the form
`a_ 1 x + b_ 1 y + c _ 1 = 0 and a _ 2 x + b_ 2 y + c_ 2 = 0 `
where ` a_ 1 = k , b_ 1 = 3, c_ 1 = ( 3 - k ) and a_ 2 = 12, b_2 = k , c _ 2 = - k `
Let the given system of equations have infinitely many solutions.
Then, ` (a_ 1 ) /(a _ 2) = (b _ 1 ) /( b _ 2 ) = (c_1 ) /( c_ 2 ) `
` rArr (k )/( 12) = ( 3) /(k ) = (( 3- k ))/( - k ) `
`rArr (k)/( 12) = ( 3) /(k ) = ( k - 3 ) /( k )`
` rArr ( k ) /(12) = ( 3) /(k ) and ( 3 ) /( k ) = ( k - 3 ) /( k ) `
`rArr k ^(2) = 36 and k ^(2) - 6k = 0 `
` rArr ( k = 6 or k =- 6) and k (k - 6) = 0 `
` rArr ( k = 6 or k = - 6) and ( k = 0 or k = 6 )`
`rArr k = 6 `
Hence, the given system of equations has infinitely many solutions when ` k = 6 `.