What is the solution for the variables?

6x + 3y - 7z = 7

x - 2y + 7z = 10

-3x + 2y - 8z = -16

7x + y = 17

8x - 16y + 56z = 80

-21x + 14y - 56z = -112

-13x - 2y = -32

14x + 2y = 34

x = 2, y = 3, and z = 2

6x + 3y - 7z =...7...(1).;

  x...-2y +7z =  10...(2).;

3x...+2y -8z = -16...(3).;

[(1)+(2)] --->........7x + y = 17...(4).;

[8(2)+7(3)] --->..29x -2y =-32...(5).; 

[2(4)+(5)] --->....43x = 2, ie., x = (2/43). Then (4) ---> y = 17 - 7(2/43) = 17 - (14/43);

= (17*43 -14)/43 = (717/43).;

[(2)+(3)] ---> 4x-z=-6, ie., z = 4x+6 = 4(2/43)+6 =(8/43)+6 = (6*43 +8)/43 = (266/43);

Then [x , y , z] = [(2/43) , (717/43) , (266/43)]. 

The solution is :  x= -142/104, y = -395/104 , z= 91/104

6x + 3y - 7z = 7 and x - 2y + 7z = 10 and 3x + 2y - 8z = -16

Let's start with solving the second equation for x in terms of y and z, then substitute into the other two equations:

x - 2y + 7z = 10

x = 2y - 7z + 10

6x + 3y - 7z = 7 and 3x + 2y - 8z = -16

6(2y - 7z + 10) + 3y - 7z = 7 and 3(2y - 7z + 10) + 2y - 8z = -16

12y - 42z + 60 + 3y - 7z = 7 and 6y - 21z + 30 + 2y - 8z = -16

15y - 49z = -53 and 8y - 29z = -46

Now I'll solve the last equation for y in terms of z and substitute into the other:

8y - 29z = -46

8y = 29z - 46

y = (29z - 46) / 8

15y - 49z = -53

15(29z - 46) / 8 - 49z = -53

I'll start by multiplying both sides by 8:

15(29z - 46) - 392z = -424

435z - 690 - 392z = -424

43z = 266

z = 266/43

Now we have z, we can work back to solve for y, then x:

8y = 29z - 46

8y = 29(266/43) - 46

8y = 7714/43 - 46

8y = 7714/43 - 1978/43

8y = 5736/43

y = 717/43

x = 2y - 7z + 10

x = 2(717/43) - 7(266/43) + 10

x = 1434/43 - 1862/43 + 10

x = -428/43 + 10

x = -428/43 + 430/43

x = 2/43

The solution is:

x = 2/43 and y = 717/43 and z = 266/43

6x + 3y - 7z + x - 2y + 7z = 7 + 10

7x + y + 0z = 17

7x + y = 17

y = 17 - 7x

8 * (x - 2y + 7z) + 7 * (3x + 2y - 8z) = 8 * 10 + 7 * (-16)

8x - 16y + 56z + 21x + 14y - 56z = 80 - 112

29x - 2y = -32

29x - 2 * (17 - 7x) = -32

29x - 34 + 14x = -32

43x = 2

x = 2/43

y = 17 - 7x

y = 17 - 14/43

y = (17 * 43 - 14) / 43

y = (680 + 51 - 14) / 43

y = (680 + 37) / 43

y = 717/43

x - 2y + 7z = 10

2/43 - 1434/43 + 7z = 10

-1432/43 + 7z = 10

7z = 430/43 + 1432/43

7z = 1862/43

7z = (1400 + 420 + 42) / 43

z = (200 + 60 + 6) / 43

z = 266/43