-x -3y +4z = 3
5x -8y +5z = 27
5x -2y +6z = 1
0 is neither + nor -, but I used +0 below to make the matrix line up better.
The solution below looks a bit messy because fractions and exact numbers are being used. If you use decimals the matrix will look prettier and won't have as many big numbers, but the answer will be slightly off due to rounding errors.
matrix
-1 -3 +4 | +3
+5 -8 +5 | +27
+5 -2 +6 | +1
on the third row, subtract out the 2nd row
-1 -3 +4 | +3
+5 -8 +5 | +27
+0 +6 +1 | -26
multiply the top row by 5
-5 -15 +20 | +15
+5 -8 +5 | +27
+0 +6 +1 | -26
in the top row, add the bottom row
+0 -23 +25 | +42
+5 -8 +5 | +27
+0 +6 +1 | -26
multiply the bottom row by -25
+0 -23 +25 | +42
+5 -8 +5 | +27
+0 -150 -25 | +650
in the bottom row, add the top row
+0 -23 +25 | +42
+5 -8 +5 | +27
+0 -127 +0 | +692
multiply the top row by -127
+0 +2921 -3175 | -5334
+5 -8 +5 | +27
+0 -127 +0 | +692
multiply the bottom row by 23
+0 +2921 -3175 | -5334
+5 -8 +5 | +27
+0 -2921 +0 | +15916
add the bottom row to the top row
+0 +0 -3175 | +10582
+5 -8 +5 | +27
+0 -2921 +0 | +15916
multiply the middle row by -2921
+0 +0 -3175 | +10582
-14605 +23368 -14605 | -78867
+0 -2921 +0 | +15916
multiply the bottom row by 8
+0 +0 -3175 | +10582
-14605 +23368 -14605 | -78867
+0 -23368 +0 | +127328
add the bottom row to the middle row
+0 +0 -3175 | +10582
-14605 +0 -14605 | +48461
+0 -23368 +0 | +127328
multiply the top equation by -14605
+0 +0 +46370875 | -154550110
-14605 +0 -14605 | +48461
+0 -23368 +0 | +127328
multiply the middle equation by 3175
+0 +0 +46370875 | -154550110
+46370875 +0 +46370875 | +153863675
+0 -23368 +0 | +127328
add the top equation to the middle equation
+0 +0 +46370875 | -154550110
+46370875 +0 +0 | -686435
+0 -23368 +0 | +127328
divide the top equation by 46370875
+0 +0 +1 | -154550110/46370875
+46370875 +0 +0 | -686435
+0 -23368 +0 | +127328
divide the middle equation by 46370875
+0 +0 +1 | -154550110/46370875
+1 +0 +0 | -686435/46370875
+0 -23368 +0 | +127328
divide the bottom equation by -23368
+0 +0 +1 | -154550110/46370875
+1 +0 +0 | -686435/46370875
+0 +1 +0 | -127328/23368
put the middle equation on top and the top equation on bottom
+1 +0 +0 | -686435/46370875
+0 +1 +0 | -127328/23368
+0 +0 +1 | -154550110/46370875
that's an answer, but the fractions on the right can be reduced.
factor a 5 out from the top and bottom equations and a 4 out from the middle equation
+1 +0 +0 | -137287/9274175
+0 +1 +0 | -15916/2921
+0 +0 +1 | -30910022/9274175
factor a 23 out of all three equations
+1 +0 +0 | -5969/403225
+0 +1 +0 | -692/127
+0 +0 +1 | -1343914/403225
factor a 127 out of the top and bottom equations
+1 +0 +0 | -47/3175
+0 +1 +0 | -692/127
+0 +0 +1 | -10582/3175
Answer: x = -47/3175, y = -692/127, z = -10582/3175
or, in decimal rounded to 4 places after the decimal
Answer: x = -0.0418, y = -5.4488, z = -3.3329