`2x + y - z = 7, x - 2y + 2z = -9, 3x - y + z = 5` Solve the system of linear equations and check any solutions algebraically.

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.