2v +  w + 2x - 2y +  z = 10
 v + 2w + 3x + 3y - 3z = -6
 v -  w - 2x - 2y + 3z =  8
2v - 2w +  x - 3y + 2z = 15
3v + 3w -  x +  y -  z =  1

It's too much to ask of any student to solve this
by hand and without technology.  So I'll do it using
a TI-84.  If your teacher doesn't allow this, then he
or she is unreasonable and should be fired.

Make the augmented matrix



Put that in your TI-84 by pressing 2ND x-1

Highlight EDIT
Make it read 5 x6 and enter the 30 matrix numbers left to
right, top to bottom

Press 2ND MODE
Press CLEAR
Press 2ND x-1
Highlight MATH
Scroll to highlight B:rref(
Press ENTER

   Screen will read rref(
Press 2ND x-1
Press ENTER
Press ) to close the parentheses
   Screen will read rref([A])
Press MATH ENTER ENTER

Screen reads



which means

v = 178/75
w = -71/75
x =  14/25
y = -59/25
z =   9/25 

I'll say again, if your teacher said to do that by hand and
not use a calculator or technology, then he or she is totally
unreasonable and should be fired.

Edwin