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Question
Discuss the nature of solutions of the following system of equations
2y + z = 3(– x + 1); – x + 3y – z = – 4; 3x + 2y + z = `-1/2`
Solution
2y + z = 3 (– x + 1)
2y + z = – 3x + 3 ...(1)
3x + 2y + z = `-1/2`
– x + 3y – z = – 4
x – 3y + z = 4 ...(2)
3x + 2y + z = `-1/2`
Multiply by 2 6x + 4y + 2z = – 1 ....(3)
3x + 2y + z = 3 ....(1)
x – 3y + z = 4 ....(2)
(–) (+) (–) (–)
(1) – (2) ⇒ 2x + 5y + 0 = – 1
2x + 5y = – 1 ....(4)
(1) × 2 ⇒ 6x + 4y + 2z = 6 ....(1)
(3) × 1 ⇒ 6x + 4y + 2z = – 1 ....(3)
(–) (–) (–) (+)
(1) – (3) ⇒ 0 + 0 + 0 = 7
Hence we arrive at a contradiction as 0 ≠ 7
This means that the system is inconsistent and has no solution.
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