consistent and inconsistent systems with 3 variables

Question

Find wether the following systems are consistent or not:
2x-y+z=4
3x-y+z=6
4x-y+2z=7
-x+y-z=9

Answer 1

The answer is no.

 

The only difference between the first two equations is the coefficient of x goes up by 1 while the answer goes up by 2.  Therefore, if the systems are consistent, x must equal 2. 

 

Now, if you substitute that back into either of the first two equations, - y + z must equal 0.  Therefore, y = z.

 

Looking at the third equation, the coefficients of both x and z go up by 1 while the answer goes down by 1.  Since increasing the coefficient of x would cause the answer to go up by 2 (if it were consistent) and the answer only went up by one, then z (and hence y) must equal -1.

 

Now, looking at the final equation, if you substitute 2 for x and -1 for y and z, you would get an answer of -2 which does not equal 9.  Therefore, the systems are not consistent.