4.4.34. For each of the following linear systems, (i) verify compatibility using the Fredholm alternative, (ii) find the general solution, and (iii) find the solution of minimum Euclidean norm: (a) 2x - 4y = -6, -x + 2y = 3, (b) 2x + 3y = -1, 3x + 7y = 1, -3x + 2y = 8, (c) 6x - 3y + 9z = 12, 2x - y + 3z = 4, (d) x + 3y + 5z = 3, -x + 4y + 9z = 11, 2x + 3y + 4z = 0, (e) x_1 - 3x_2 + 7x_3 = -8, 2x_1 + x_2 = 5, 4x_1 - 3x_2 + 10x_3 = -5, -2x_1 + 2x_2 - 6x_3 = 4. (f) x - y + 2z + 3w = 5, 3x - 3y + 5z + 7w = 13, -2x + 2y + z + 4w = 0.
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(i) The given system of equations is: 2x + 3y = -1 2x + 4y = -6 6x - 3y + 92 = 12 This system is inconsistent and has no solution because the first two equations imply that y = -2x - 1/3 and y = -2x - 3/2, which are contradictory. Show more…
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Solve the following systems of equations by Cramer's rule. 1. x + 2y = -3, 3x + y = 1. 2. 2x + y = 3, x - y = 6. 3. 4a + 2b = 0, 5a - 3b = 10. 4. 3s - 4t = 30, -2s + 3t = -10. 5. 2x - 8y = 200, -x + 4y = 150. 6. x + y - 2z = 3, 2x - y + 3z = 2. 7. x + y = 15, x + z = 15, y + z = 10. 8. 3x + y + z = 4, x - y + 2z = 15, 2x - 2y - z = 5. 9. x + 2y - 2z = -1, 2x + y + z = 5, -x + y - z = -2. 10. 2a + 3b - c = 4, -a - 2b + c = -2, 3a - b = 2. 11. 2x + 3y + 2z = 3, 3x + y + 5z = 2, 7y - 4z = 5. 12. 5r + 8s + t = 2, 2s + t = -1, 4r + 3s - t = 3. 13. x + 2y + z + w = 7, 3x + 4y - 2z - 4w = 13, 2x + y - z + w = -4, x - 3y + 4z + 5w = 0.
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Consider the following linear system: x + y + z = 3 (S) 2x + 3y + 4z = 11 4x + 9y + 16z = 41 The augmented matrix corresponding to (S) is: [1 1 1 | 3] [2 3 4 | 11] [4 9 16 | 41] With reference to Exercise and by using the Gaussian elimination method, the solution of (S) is: x = 2 + 2t y = 2 - t z = t
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