1. Find all solutions of the following systems. For systems with two or three variables describe the
solution set geometrically if you can. x1 + 2x2 - x3 + 2x4 = 1 2x1 - x2 + 2x3 - x4 = 2 3x1 + x2 +
x3 + x4 = 3 -x1 + 3x2 - 3x3 + 3x4 = -1
Solution
x 1 + 2x 2 - 3x 3 + 2x 4 = 1 .................i
2x 1 - x 2 + 2x 3 - x 4 = 2 ..................ii
3x 1 + x 2 + x 3 + x 4 = 3..................iii
-x 1 + 3x 2 - 3x 3 + 3x 4 = -1.................iv
multiplying eq ii by 2 and adding in eq i
x 1 + 2x 2 - 3x 3 + 2x 4 = 1
4x 1 - 2x 2 + 4x 3 - 2x 4 = 4
5x 1 + x 3 = 5 .........................v
multiplying eq iii by 3 and subtracting from eq iv
-x 1 + 3x 2 - 3x 3 + 3x 4 = -1
-9x 1 -3x 2 - 3x 3 - 3x 4 = -9
-10x 1 - 6x 3 = -10 ....................vi
solving v and vi
multiplying eq v by 2 and adding in vi
-10x 1 - 6x 3 = -10

2. 10x 1 + 2x 3 = 10
this gives x 3 = 0
x1 = 1 .......................