Final answer:
The student's question is about finding conditions on 'a' and 'b' in a system of linear equations to determine when there are no, one, or infinitely many solutions. For no solutions, the system must be inconsistent; for one solution, the equations must all intersect at one point, and for infinitely many solutions, the equations must be dependent.
Step-by-step explanation:
The student question concerns a system of linear equations and how to determine the number of solutions (none, exactly one, three, or infinitely many). The system in question is:
- x - y + 2z = 4
- 3x - 2y + 9z = 14
- 2x - 4y + az = b
To determine the number of solutions, we examine the coefficients of the given equations and look for conditions on a and b that correspond to each case (a through d).
For no solution, the system must be inconsistent, which generally happens when two lines are parallel but with different intercepts. For one solution, the system's equations must intersect at a single point. For exactly three solutions, it is not possible as a system of linear equations in three variables can't have exactly three solutions; it will always have either no solution, exactly one solution, or infinitely many solutions. For infinitely many solutions, the equations must be dependent, essentially describing the same plane in three-dimensional space.
To find specific values of a and b, we use elimination or substitution to simplify the system and uncover the necessary conditions for each case.