Question

Find values of a, b, and c (if possible) such that the system of linear equations has a unique solution, no solution, and infinitely many solutions. (if not possible, enter impossible.) x y = 2 y z = 2 x z = 2 ax by cz = 0

Answer 1

The system has a unique solution when a = 1, b = -1, and c = 1. It has no solution when a = 0, b = 0, and c = 0 and it has infinitely many solutions when a = 1, b = 1, and c = 2.

Step-by-step explanation:

To determine the values of a, b, and c such that the system of linear equations has a unique solution, no solution, or infinitely many solutions, we can analyze the given equations.

We have three equations:

1. x + y = 2

2. y + z = 2

3. x + z = 2

For the system to have a unique solution, we need the determinant of the coefficients to be non-zero. If the determinant is zero, the system will have no solution or infinitely many solutions.

By solving the equations using substitution or elimination, we find that the system has a unique solution when a = 1, b = -1, and c = 1.

It has no solution when a = 0, b = 0, and c = 0.

And it has infinitely many solutions when a = 1, b = 1, and c = 2.