Consider the system of linear equations: x + y + z = x + y + z = x + y + z = 1 If the system has infinitely many solutions and [0, 4 ] , then the sum of the values of 4 for all valid is equal to :

Step-by-step solution

For the system to have infinitely many solutions, the determinant of the coefficient matrix must be zero. = vmatrix & & 1 & & 1 1 & 1 & 1 vmatrix = 0 Expanding along the first row: ( - 1) - ( - 1) + 1( - ) = 0 ^2 - - ^2 + + - = 0 ( ^2 - ^2 ) - 2( - ) = 0 ( - )( + ) - 2( - ) = 0 ( - )( + - 2) = 0 Since the maximum value of + is 2 , the factor ( + - 2) can never be zero for any real . Therefore, we must have: - = 0 = 1 For = 1 , it is easy to verify that _x = _y = _z = 0 , confirming infinitely many solutions. The so