Question

Consider the following system of equations: ax+y+4z=22x+y+a2z=2x−3z=a​ a) For what values of a does the system have one, none, or infinitely many solutions?

b) Replace the right-hand sides of the system by b1​,b2​, and b3​, respectively. FInd a necessary and sufficient condition fort he new system of equations to have infinitely many solutions.

Answer 1

To determine the values of a for which the system of equations has one, none, or infinitely many solutions, we need to consider the determinant of the coefficient matrix.

Step-by-step explanation:

To determine the values of a for which the system of equations has one, none, or infinitely many solutions, we need to consider the determinant of the coefficient matrix. The determinant is calculated as follows:

| a 1 4 |

| 2 1 a2 |

| 1 -3 0 |

The system has one solution if and only if the determinant is nonzero. It has infinitely many solutions if and only if the determinant is zero and at least one column is all zeros. Otherwise, it has no solution.

For part b, if we replace the right-hand sides of the system by b1, b2, and b3 respectively, then the new system of equations will have infinitely many solutions if and only if the determinant of the coefficient matrix is zero and the determinant of the augmented matrix with the new right-hand sides is also zero.