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Jan. 31, 2023•0 likes•4 views

Jan. 31, 2023•0 likes•4 views

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

- 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 .......................