Final Answer:
(i) Unique solution: (a) 3x + 2y = -3, x + y - 2z = -3, 2y + 17z - 5w = -50
(ii) Infinitely many solutions: (d) 3x + y + 6z + 11w = -16
(iii) No solution: (f) 2x - 5y - 4z = -13
Step-by-step explanation:
In the given systems of linear equations, the system (a) 3x + 2y = -3, x + y - 2z = -3, 2y + 17z - 5w = -50 has a unique solution. To determine this, we can use methods like substitution or elimination to solve for each variable. The system (a) is consistent and independent, meaning it has a single solution point in the solution space.
On the other hand, the system (d) 3x + y + 6z + 11w = -16 has infinitely many solutions. This occurs when there are dependent equations or when the number of equations is less than the number of variables. In this case, there are free variables that can take on any value, leading to an infinite set of solutions.
Lastly, the system (f) 2x - 5y - 4z = -13 has no solution. This happens when the system is inconsistent, indicating that the planes represented by the equations do not intersect at any point. In geometric terms, these planes are parallel and never meet.
To summarize, the unique solution arises from consistent and independent equations, infinite solutions result from dependent equations or fewer equations than variables, and no solution occurs when the system is inconsistent. These classifications provide insights into the geometric relationships between the planes represented by the equations.
Complete Question Here:
Which of the following systems has (i) a unique solution? (ii) infinitely many solutions? (iii) no solution? In each case, find all solutions: (a) 3x + 2y =-3 x+y-2z=-3, +2y17z -5w-50, (e) 3x+y+6z +11w- 16, (f) 2x-5y -4z-13,