Precalculus, Chapter 7, 7.3, Section 7.3, Problem 29

You may use the reduction method to solve the system, hence, you may multiply the first equation by 2, such that:
2(2x + y - z) = 2* 7
4x + 2y - 2z = 14
You may now add the equation 4x + 2y - 2z = 14 to the second equation x - 2y + 2z = - 9, such that:
4x + 2y - 2z + x - 2y + 2z= 14 - 9
5x = 5 => x = 1
You may replace 1 for x in equation 3x - y + z = 5 , such that:
3 - y + z = 5 => -y + z = 2
You may also replace 1 for x in equation x - 2y + 2z = -9 , such that:
1 - 2y + 2z = -9 => - 2y + 2z = -10 => y - z = 5
You may add the equations y - z = 5 and -y + z = 2:
-y + z + y - z= 2 + 5 => 0 = 7 inaccurate.
Hence, evaluating the solution to the given system, yields that there are no solutions.