please solve using the augmented matrix

-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
by Top Rated User (103k points)