Question 189199
Solve each of these system of equations. 
1) x + 2y + z = 10 
2x - y + 3z = -5 
2x - 3y - 5z = 27


a) x = 4, y = 7, z = -5
b) x = -5, y = 4, z =7
c) x = 7, y = -5, z = 4
d) x = 7, y = 4, z = -5 
--------------------
 x + 2y + z = 10 
2x - y + 3z = -5 
2x - 3y - 5z = 27
There's more than one way to do these.  If you get one on a timed test, and you have 4 solutions to choose from, as in this one, sub the values in to find which of the 4 it is.  That's quicker by far, IF you don't have to show your work for the test.
Then there's elimination and substitution.  You pick one of the 3 variables and eliminate it from 2 of the equations.  x is as good as any for this problem.
Multiply by whatever is needed to get the same coeffs for all 3 eqns, in this case eqn 1 times 2 will do it.

 x + 2y + z = 10 -->
2x + 4y + 2z = 20
2x - y + 3z = -5 
2x - 3y - 5z = 27
Then subtract one eqn, the 3rd, from the 1st and the 2nd.
2x + 4y + 2z = 20
2x - 3y - 5z = 27
----------------- Subtract
0x + 7y + 7z = -7  eqn A
--------------
2x - y + 3z = -5 
2x - 3y - 5z = 27
----------------- Subtract
0x + 2y + 8z = -32 eqn B
Now there are 2 eqns in 2 variables, A and B.
Multiply again to get the same coeffs
y + z = -1
y + 4z = -16
------------ Subtract
0y -3z = 15
z = -5 zzzzzzzzzzzzzzzzzzzzzzzzzzzz
------------
y + z = -1
y = 4 yyyyyyyyyyyyyyyyyyyyyyyyyyyyyy
------------
x + 2y + z = 10
x = 7 xxxxxxxxxxxxxxxxxxxxxxxxxxxx
That's answer d.
--------------------
Another method is determinants.  I made an Excel sheet that does these up to 4 by 4.
-------------------
I don't have a calculator that does these.