Find the value of k for which the following system of linear equations has infinite solutions: x + [k + 1] y = 5 and [k + 1]x + 9y = 8k - 1
The value of k for which the following system of linear equations has infinite solutions x + [k + 1]y = 5, [k + 1]x + 9y = 8k - 1 is 2. First-order equations include linear equations. In the coordinate system, linear equations are defined for lines.
Solution : The given system of equations is
` x + [ k + 1 ] y - 5 = 0 " " `… [i]
`[k + 1] x + 9y + [1 - 8k ] = 0 " " `… [ii]
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 = 1 , b_ 1 = [ k + 1 ], c_ 1 = - 5`
and ` a_ 2 = [ k + 1 ], b_ 2 = 9, c_ 2 = [ 1 - 8k ]`.
` therefore [ a_ 1 ] /[ a_ 2] = [1]/[[ k +1]] , [b _ 1 ] /[ b_ 2 ] = [[ k + 1 ] ]/[9 ] and [c _ 1 ] /[c _ 2] = [-5]/[[ 1- 8k ]] = [ 5]/[[ 8k - 1 ]]`
Let the given system of equations have infinitely many solutions.
Then, ` [a_ 1 ]/[a_ 2 ] = [b_ 1]/[b_ 2 ] = [c_ 1 ]/[c_ 2]`
` rArr [ 1]/[[ k + 1 ]] = [[k + 1 ]]/[ 9] = [ 5]/[[ 8 k - 1 ]] `
` rArr [1]/[[ k + 1 ]] = [[k + 1 ] ]/[9] and [[ k +1 ]]/[ 9 ] = [ 5]/[ [ 8 k - 1]]`
` rArr [ k + 1 ] ^[2] = 9 and [ k + 1 ] [ 8 k - 1 ] = 45`
` rArr [ k + 1 = 3 or k + 1 = - 3 ] and 8 k ^[2] + 7k - 46 = 0 `
` rArr [ k = 2 or k = - 4 ] and [ k - 2 ] [ 8k + 23]= 0 `
` rArr [ k = 2 or k = - 4] and [ k = 2 or k = [-23]/[8]] `
`rArr k = 2`.
Hence, the given system of equations will have infinitely many solutions when k = 2.