System of Linear Equations-Exercise 23.2-Solved

1.

Please copy the question from book.

a.

Soln:

From the given equations,

Or, x1 = 13 (-9 – 2x2),x2 = 13(2x1 + 6)

Initially, x1 = 0, x2 = 0.

Iteration: I.

Or, x1 = 13(-9 – 0) = -3.

Or, x2 = 13(- 6 + 6) = 0

Iteration II,

Or, x1 = 13(-9 – 0) = -3.

Or, x2 = 13(-6 + 6) = 0

So, the solution is x1 = -3, x2 = 0.

b.

Soln:

From the given equation,

Or, x1 = 15(- 2 + 2x2), x2 = 15(24 – 2x1).

Initially, x1 = 0,x2 = 0.

Iteration I

Or, x1 = 15(-2 + 0) = −25 = -0.4

Or, x2 = 
$\frac{\mathrm{1/}}{\mathrm{5*}}\left(24+2\ast \frac{\mathrm{2/}}{5}\right)$ = 12425 = 4.96.

Iteration II

Or, x1 = 15(-2 + 2x2) = 15(-2 + 2 * 4.96) = 1.584.

Or, x2 = 15(24 – 2 * 1.584) = 4.166.

c.

Soln:

From the given equation,

Or, x1 = 13(- 1 – x2 – 2x3), x2 = 13(5 – 2x1 – x3), x3 = x1 + 2x2 – 8.

Initially, x1 = 0,x2 = 0, x3 = 0.

Iteration I

Or, x1 = 13(- 1 – 0 – 0) = -0.333

Or, x2 = 13(5 + 2 * 0.333 – 0) = 1.889.

Or, x3 = -0.333 + 2 * 1.889 – 8 = -4.555

Iteration II:

Or, x1 = 13(- 1 – 1.889 + 2 * 4.555) = 2.074.

Or, x2 = 13(5 – 2 * 2.074 + 4.555) = 1.802

Or, x3 = 2.074 + 2 * 1.802 – 8 = -2.322

d.

Soln:

From the given equations,

Or, x1 = 12(4 – 3x2 – x3), x2 = 12(3x1 – 2x3 – 4), x3  = 19(3 + 3x1 – 8x2).

Initially, x1 = 0, x2 = 0, x3 = 0.

Iteration : I

Or, x1 = 12 (4 – 0 – 0) = 2.

Or, x2 = 12 (3 * 2 – 0 – 4) = 1.

Or, x3 = 19(3 + 3 * 2 – 8 * 1) = 0.1111

Iteration: II

Or, x1 = 12(4 – 3 * 1 – 0.1111) = 0.444

Or, x2 = 12 (3 * 0.444 – 2 * 0.1111 – 4) = -1.445.

Or, x3 = 19 (3 + 3 * 0.444  + 8 * 1.445) = 1.766.

2.

Please copy the question from book.

a.

Soln:

From the given equations,

Or, x1 = 13(5 – x2) and x2 = 13(5 – x1)
Initially, x1 = 0,x2 = 0.

Iteration: I

Or, x1 = 53 = 1.667

Or, x2 = 12(5 – 1.667) = 1.6665

Iteration: II

Or, x1 = 13(5 – 1.6665) = 1.1112

Or, x2 = 13(5 – 1.1112) = 1.9444

Iteration: III

Or, x1 = 13(5 – 1.9444) = 1.018.

Or, x2 = 12(5 – 1.018) = 1.991.

Iteration: IV

Or, x1 = 13(5 – 1.991) = 1.003

Or, x2 = 13(5 – 1.003) = 1.9985.

Iteration: V

Or, x1 = 13(5 – 1.9985) = 1.0000

Or, x2 = 12(5 – 1.0000) = 2.

So, The solution is x1 = 1,x2 = 2.

b.

Soln:

From the given equations,

Or, x1 = 15(7 + 2x2), x2 = 13(18 – 2x1)

Initially, x1 = 0, x2 = 0.

Iteration: I

Or, x1 = 15(7 + 0) = 1.4

Or, x2 = 13(18 – 2 * 1.4) = 5.0667.

Iteration: II

Or, x1 = 15(7 + 2 * 5.0667) = 3.42668

Or, x2 = 13(18 – 2 * 3.42668) = 3.71555

Iteration: III

Or, x1 = 15(7 + 2 * 3.42668) = 2.7707

Or, x2 = 13(18 – 2 * 2.7707) = 4.1529.

Iteration: IV

Or, x1 = 15(7 + 2 * 2.7707) = 2.50828

Or, x2 = 13(18 – 2 * 2.50828) = 4.3278

Iteration: V.

Or, x1 = 15 (7 + 2 * 4.3278) = 3.13112.

Or, x2 = 13 (18 – 2 * 3.13112) = 3.9126.

Iteration: VI.

Or, x1 = 15(7 + 2 * 3.9126) = 2.96504.

Or, x2 = 13(18 – 2 * 2.96504) = 4.0233.

Iteration: VII

Or, x1 = 15(7 + 2 * 4.0233) = 3.0093.

Or, x2 = 13(18 – 2 * 3.0093) = 3.9938.

Iteration: VIII

Or, x1 = 15(7 + 2 * 3.9938) = 2.99752

Or, x2 = 13(18 – 2 * 2.99752) = 4.

Iteration: IX

Or, x1 = 15(7 + 2 * 4) = 3.

Or, x2 = 13(18 – 2 * 3) = 4.

So, the solution is x1 = 3,x2 = 4.

c.

Soln:

From the given equations:

Or, x1 = 12(8 + x2)

Or, x2 = 17(-5 – 3x1)

Initially, x1 = 0,x2 = 0.

Iteration: I

Or, x1 = 12(8 + 0) = 4.

Or, x2 = 17(- 5 – 3 * 4) = -2.4286.

Iteration: II

Or, x1 = 12(8 – 2.4286) = 2.7857

Or, x2 = 17 (- 5 – 3 * 2.7857) = -1.90816.

Iteration: III

Or, x1 = 12(8 – 1.90816) = 3.04592.

Or, x2 = 17(-5 – 3 * 3.04592) = -2.01968

Iteration: IV.

Or, x1 = 12(8 – 2.01968) = 2.99016.

Or, x2 = 17(- 5 – 3 * 2.99016) = -1.99578.

Iteration: V.

Or, x1 = 12(8 – 1.99578) = 3.

Or, x2 = 17(- 5 – 3 * 3) = -2.

So, the solution is x1 = 3, x2 = -2.

d.

Soln:

From the given equation,

Or, x = 15(13 + 3y)

Or, y = 14(3x – 10)

Initially, x = 0, y = 0.

Iteration: I

Or, x = 15(13 + 0) = 2.6

Or, y = 14$\frac{1}{4}$(3 * 2.6 – 10) = -0.55

Iteration II:

Or, x = 15(13 – 3 * 0.55) = 2.77

Or, y = 14(3 * 2.77 – 10) = -0.7975.

Iteration: III

Or, x = 15(13 – 3 * 0.7975) = 2.1215.

Or, y = 14(3 * 2.1215 – 10) = -0.9089.

Iteration: IV

Or, x= 15(13 – 3 * 0.9089) = 2.05466.

Or, y = 14(3 * 2.05466 – 10) = -0.959

Iteration: V

Or, x = 15(13 – 3 * 0.959) = 2.0246.

Or, y = 14(3 * 2.026 – 10) = -0.98955

Iteration: VI

Or, x = 15(13 – 0.98155) = 2.01107

Or, y = 14(3 * 2.01107 – 10) = -1.

Iteration: VII

Or, x = 15(13 – 3 * 1) = 2.

Or, y = 14(3 * 2 – 10) = -1.

So, the solution is x = 2, y = -1.

3.

Please copy the question from book.

a.

Soln:

From the given equations,

Or, x1 = 13(5 – x2 – x3)

Or, x2 = 14(x1 + x3 + 2)

Or, x3 = 13(x1 + x2 + 1)

Initially, x1 = 0, x2 = 0, x3 = 0,

Iteration: I

Or, x1 = 53 = 1.667.

Or, x2 = 14(1.667 + 0 + 2) = 0.91675.

Or, x3 = 13(1.667 + 0.91675 + 1) = 1.1946.

Iteration: II

Or, x1 = 13(5 – 0.91675 – 1.1946 = 0.9629.

Or, x2 = 14(0.9629 + 0.91675 + 2) = 0.9701.

Or,x3 = 13(0.9629 + 0.9701 + 1) = 0.9777.

Iteration: III

Or, x1 = 13(5 – 0.9701 – 0.9777) = 0.0174.

Or, x2 = 14(1.0174 + 0.9777 + 2) = 0.9988

Iteration: IV

Or, x1 = 13(5 – 0.9988 – 1.0054) = 0.9986

Or, x2 = 14(0.9986 + 1.0054 + 2) = 1.

Or, x3 = 13(0.9986 + 1 + 1) = 1.

Iteration: V

Or, x1 = 13= (5 – 1 – 1) = 1.

Or, x2 = 14(1 + 1 + 2) = 1.

Or, x3 = 13(1 + 1 + 1) = 1.

So, the solution is x1 = 1, x2 = 1, x3 = 1.

b.

Soln:

From the given equations,

Or, x1 = 15(15 – 3x2 + x3)

Or, x2 = 14(2x1 + x3 + 4)

Or, x3 = 13(x1 – x2 + 13)

Initially, x1 = 0, x2 = 0, x3 = 0.

Iteration: I

Or, x1 = 15(15 – 0 + 0) = 3.

Or, x2 = 14(2 * 3 + 0 + 4) = 2.5

Or, x3 = 13(3 – 2.5 + 13) = 4.5.

Iteration: II

Or, x1 = 15(15 – 3 * 2.5 + 4.5) = 2.4

Or, x2 = 14(2 * 2.4 + 4.5 + 4) = 3.325.

Or, x3 = 13(2.4 – 3.325 + 13) = 4.025.

Iteration: III

Or, x1 = 15(15 – 3 * 3.325 + 4.025) = 1.81.

Or, x2 = 14(2 * 1.81 + 4.025 + 4) = 2.9112.

Or, x3 = 13(1.81 – 2.9112 + 13) = 3.9663.

Iteration IV:

Or, x1 = 15(15 – 3 * 2.9112 + 3.9663) = 2.0465.

Or, x2 = 14(2 * 2.0465 + 3.9663 + 4) = 3.0148.

Or, x3 = 13(2.0465 – 3.0148 + 13) = 4.0106.

Iteration: V

Or, x1 = 15(15 – 3 * 3.0148 + 4.0106) = 1.9934.

Or, x2 = 14(2 * 1.9934 + 4.0106 + 4) = 2.99935.

Or, x3 = 13(1.9934 – 2.99935 + 13) = 3.9980.

Iteration: VI

Or, x1 = 15(15 – 3 * 2.99935 + 3.9980) = 2.

Or, x2 = 14(2 * 2 + 3.9980 + 4) = 3.

Or, x3 = 13(2 – 3 + 13) = 4.

So, the solution is x1 = 2, x2 = 3, x3 = 4.

c.

Soln:

From the given equations:

Or, x1 = 14(7 – x2 – x3)

Or, x2 = 15(2x1 + 2x3 – 1)

Or, x3 = 13(6 – x1 + x2)

Initially, x1 = 0, x2 = 0, x3 = 0.

Iteration: I

Or, x1 = 14(7 – 0 – 0) = 1.75.

Or, x2 = 15(2 * 1.75 + 0 – 1) = 0.5

Or, x3 = 13(6 – 1.75 + 0.5) = 1.5833.

Iteration: II

Or, x1 = 14(7 – 0.5 – 1.5833) = 1.2292.

Or, x2 = 15(2 * 1.2292 + 2 * 1.5833 – 1) = 0.9250.

Or, x3 = 13(6 – 1.2292 + 0.9250) = 1.8986.

Iteration: III

Or, x1 = 14(7 – 0.9250 – 1.8986) = 1.0441.

Or, x2 = 15(2 * 1.0441 + 2 * 1.8986 – 1) = 0.9771

Or, x3 = 13(6 – 1.0441 + 0.9771) = 1.9777

Iteration: IV

Or, x1 = 14(7 – 0.9771 – 1.9777) = 1.0113.

Or, x2 = 15(2 * 1.0113 + 2 * 1.9777 – 1) = 0.9956.

Or, x3 = 13(6 – 1.0113 + 0.9956) = 1.9948

Iteration: V

Or, x1 = 14(7 – 0.9956 – 1.9948) = 1.

Or, x2 = 15(2 * 1 + 2 * 1.9948 – 1) = 1.

Or, x3 = 13(6 – 1 + 1) = 2.

So, the solution is x1 = 1, x2= = 1 and x3 = 2.

d.

Soln:

From the given equations,

Or, x = 13(2 – y + z).

Or, y = 15(2x + z – 20).

Or, z = 18(x – 3y – 3).

Initially, x = 0, y = 0, z = 0,

Iteration: I

Or, x = 13(2 – 0 + 0) = 0.6667.

Or, y = 15(2 * 0.6667 + 0 – 20) = -3.7333.

Or, z = 18(0.6667 + 3 * 3.7333 – 3) = 1.1083.

Iteration: II

Or, x = 13(2 + 3.7333 + 1.1083) = 2.2805.

Or, y = 15 (2 * 2.2805 + 1.1083 – 20) = -2.8661

Or, z = 18(2.2805 + 3 * 2.8661 – 3) = 0.9848.

Iteration: III

Or, x = 13(2 + 2.8661 + 0.9848) = 1.9503.

Or, y = 15(2 * 1.9503 + 0.9848 – 20) = -3.0229.

Or, z = 18$\frac{1}{8}$(1.9503 + 3 * 3.0229 – 3) = 1.

Iteration: IV.

Or, x = 13(2 + 3.02229 + 1) = 2.

Or, y = 15(2 * 2 + 1 – 20) = -3.

Or, z = 18 (2 + 3 *3 – 3) = 1.

So, the solution ix x = 2, y = -3, z = 1.