Final answer:
The existence and number of solutions for the given system of equations depend on the value of 't'. The system could have no solution, one solution, or many solutions. However, without specific information on the value of 't', it's impossible to definitively determine the number of solutions.
Step-by-step explanation:
The student's question involves a system of three equations with three variables (x, y, and z), where t is a parameter within the system. Solving this kind of system involves checking solutions for consistency across all equations. Since the question involves a parameter, the existence and number of solutions will depend on the value of 't'. There are a few possibilities for the system:
- No solution: This would occur if the planes represented by the equations are parallel and do not intersect, or if they intersect in a way that doesn't result in a consistent set of x, y, and z that satisfies all equations.
- One solution: If the three planes intersect at a single point, the system will have a unique solution representing the coordinates of the intersection.
- Many solutions: This may occur if the system represents planes that intersect along a line, which would mean there is an infinite set of points (solutions) where all three equations are satisfied.
To determine which of these scenarios applies, we would need to carry out a step-by-step algebraic process to solve the system.
Step 1: Combine equations to eliminate one variable.
Step 2: Simplify the resulting equations.
Step 3: Solve the simplified system for two variables.
Step 4: Substitute back to find all three variables.
Without specific values for 't', we cannot determine the number of solutions to this problem. Hence, we cannot select (A), (B), or (C) without additional information.