Show that the system
\[ \begin{array}{l} x _ {1} - 2 x _ {2} + x _ {3} + 2 x _ {4} = 1 \\ x _ {1} + x _ {2} - x _ {3} + x _ {4} = 2 \\ x _ {1} + 7 x _ {2} - 5 x _ {3} - x _ {4} = 3 \end{array} \]
has no solution.
Solution
1. The third equation minus \(3\) times the second equation plus \(2\) times the first equation gives \(0 = -1\).
1.1. Left-hand side: \((x_1 + 7x_2 - 5x_3 - x_4) - 3(x_1 + x_2 - x_3 + x_4) + 2(x_1 - 2x_2 + x_3 + 2x_4) = 0\).
the coefficients cancel: \(x_1 - 3x_1 + 2x_1 = 0\), \(7x_2 - 3x_2 - 4x_2 = 0\), \(-5x_3 + 3x_3 + 2x_3 = 0\), \(-x_4 - 3x_4 + 4x_4 = 0\).
1.2. Right-hand side: \(3 - 3(2) + 2(1) = 3 - 6 + 2 = -1\).
arithmetic.
2. Hence any solution would satisfy \(0 = -1\), a contradiction.
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3. Therefore the system has no solution.
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4. Q.E.D.
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