(a) Consider the following flow diagram: 40 60 x3 x2 x4 30 x5 10 At each vertex, the flow in must equal the flow out. (i) Starting from the top left vertex and reading clockwise, specify four linear equations involving a1,..., x5 and write these with all the variables on the left hand side, and constants on the right hand side. (ii) You are told 10 10 1 140 01 10 1 0 0 0 1 1 0 0 0 0 0 1 -1 80 1 1 -1 0 120 60 0 0 0 1 1 1 1 20 20 1 60 Explain why it follows from this that the reduced row echelon form of the augmented matrix corresponding to the linear system in (a) is 10 10 1 | 70 01 1 0 1 30 0 0 0 1 1 10 0 0 0 0 0| 0 (iii) Find the solution of the linear system in (i), given that it is required r3 = 0 and 25 = 10. (b) A system of three simultaneous equations in the unknowns a, y, z has only two solutions (x, y, z) = (0,0, 0) and (r, y, 2) = (1,0,0). Is it possible for these equations to be linear? Clearly state the theory being used. (c) Let k ER be given and consider the system of equations for the unknowns x, y, z specified by x + ky + 4z = 0 2x – y + 8z = 0. Calculate the solution space assuming k -.

(a) Consider the following flow diagram: 40 60 x3 x2 x4 30 x5 10 At each vertex, the flow in must equal the flow out. (i) Starting from the top left vertex and reading clockwise, specify four linear equations involving a1,..., x5 and write these with all the variables on the left hand side, and constants on the right hand side. (ii) You are told 10 10 1 140 01 10 1 0 0 0 1 1 0 0 0 0 0 1 -1 80 1 1 -1 0 120 60 0 0 0 1 1 1 1 20 20 1 60 Explain why it follows from this that the reduced row echelon form of the augmented matrix corresponding to the linear system in (a) is 10 10 1 | 70 01 1 0 1 30 0 0 0 1 1 10 0 0 0 0 0| 0 (iii) Find the solution of the linear system in (i), given that it is required r3 = 0 and 25 = 10. (b) A system of three simultaneous equations in the unknowns a, y, z has only two solutions (x, y, z) = (0,0, 0) and (r, y, 2) = (1,0,0). Is it possible for these equations to be linear? Clearly state the theory being used. (c) Let k ER be given and consider the system of equations for the unknowns x, y, z specified by x + ky + 4z = 0 2x – y + 8z = 0. Calculate the solution space assuming k -.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

(a) Consider the following flow diagram:
40
х1
60
x3
x4
30
x5
10
At each vertex, the flow in must equal the flow out.
(i) Starting from the top left vertex and reading clockwise, specify four linear equations
involving x1,...,x5 and write these with all the variables on the left hand side, and
constants on the right hand side.
(ii) You are told
0| 80
0 120
1 0 10 1
0 1 10 1
0 0 0 1 1
0 0 0 0 0
1
-1
140
1
1
–1
60
1
1
20
20
1
1
1
60
Explain why it follows from this that the reduced row echelon form of the augmented
matrix corresponding to the linear system in (a) is
1 0 10 1 | 70
0 1 1 0 1 30
0 0 0 1 1
0 0 0 0 0
10
(iii) Find the solution of the linear system in (i), given that it is required x3 = 0 and
X5 = 10.
(b) A system of three simultaneous equations in the unknowns a, y, z has only two solutions
(x, y, z) = (0,0, 0) and (æ, y, z) = (1,0, 0). Is it possible for these equations to be linear?
Clearly state the theory being used.
(c) Let k eR be given and consider the system of equations for the unknowns x, y, z specified
by
x + ky + 4z = 0
2x – y + 8z = 0.
Calculate the solution space assuming k + -.

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