Answer:
The system has an infinite solution at k = 6, otherwise for any value of k, it has zero solution.
Step-by-step explanation:
Consider the system of linear equations:
![3x_{1} -x_{2}=2\;\;\;\;\;\;(1)\\ 9x_{1} -3x_{2}=k\;\;\;\;\;(2)](https://tex.z-dn.net/?f=3x_%7B1%7D%20-x_%7B2%7D%3D2%5C%3B%5C%3B%5C%3B%5C%3B%5C%3B%5C%3B%281%29%5C%5C%209x_%7B1%7D%20-3x_%7B2%7D%3Dk%5C%3B%5C%3B%5C%3B%5C%3B%5C%3B%282%29)
The system of linear equations can have zero, one, or an infinite number of solutions:
simplify equation (1):
![x_{2}=3x_{1} -2](https://tex.z-dn.net/?f=x_%7B2%7D%3D3x_%7B1%7D%20-2)
substitute in equation (2), we get
![9x_{1} -3(3x_{1}-2)=k\\9x_{1} -9x_{1}+6=k\\0+6=k](https://tex.z-dn.net/?f=9x_%7B1%7D%20-3%283x_%7B1%7D-2%29%3Dk%5C%5C9x_%7B1%7D%20-9x_%7B1%7D%2B6%3Dk%5C%5C0%2B6%3Dk)
we cannot find the value of
and
.
so, there is no solution.
Multiply the equation (1) with 3 and put k is 6,
![3(3x_{1} -x_{2})=3\times2\\9x_{1} -3x_{2})=6](https://tex.z-dn.net/?f=3%283x_%7B1%7D%20-x_%7B2%7D%29%3D3%5Ctimes2%5C%5C9x_%7B1%7D%20-3x_%7B2%7D%29%3D6)
it means both equations are overlapped. Then, the solution has infinite solutions.
Hence, the system has an infinite solutions at k is 6 otherwise for any value of k it has no solution.