Final answer:
To find the value of k for which the system has infinitely many solutions, we solve the third equation for x, substitute into the first two equations and obtain a final equation (3 - 5k)z = 0. Setting the coefficient of z to zero, we find that k must be 3/5.
Step-by-step explanation:
The student has asked for what value of k the following system of equations has infinitely many solutions:
- kx + y + z = 0
- x + 2y + kz = 0
- -x + 3z = 0
To find this value, we must ensure that the system is consistent and dependent, which means that one equation is a scalar multiple of another, or they all represent the same plane. To ensure that this is the case, let's solve the third equation for x, which gives us x = 3z. Then let's substitute x into the first two equations:
- 3kz + y + z = 0 → y = -(3k + 1)z
- 3z + 2y + kz = 0
Substitute y from the first modified equation to the second:
- 3z + 2(-(3k + 1)z) + kz = 0
- 3z - 6kz - 2z + kz = 0
- (3 - 5k)z = 0
For the system to have infinitely many solutions, the coefficient of z must be zero, so we get 3 - 5k = 0, which means k = 3/5.