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In this task, we are required to investigate the mathematical patterns within systems of linear equations. We need to concept of matrices and algebraic equations in this task.

Extracts from this document...

Introduction

Introduction

In this task, we are required to investigate the mathematical patterns within systems of linear equations. We need to concept of matrices and algebraic equations in this task.

Body

Part A

Consider this 2×2 system of linear equationsimage00.pngimage00.png

  • Let the equations be ax + by=c,

For the first equation (image53.pngimage53.png): a = 1, b = a + 1, c = b + 1.

For the second equation ( image158.pngimage158.png): a =2, b = a – 3, c = b - 3

  • I examined the constants of two equations, I identified there is the pattern of an arithmetic sequence. For the first equation, the constants (1, 2, 3) starts with 1 and has a common difference of 1; for the second equation, the constants (2, -1, -4) starts with 2 and has a common difference of -3.
  • Then I solved the system algebraically:

image168.pngimage168.png

image177.pngimage177.png

image01.pngimage01.png

image13.pngimage13.png

image27.pngimage27.png

image37.pngimage37.png

image41.pngimage41.png

image54.pngimage54.png

image61.pngimage61.png

image74.pngimage74.png

image84.pngimage84.png

Substituting x= -1 into equation (4):image92.pngimage92.png

image105.pngimage105.png

The solution is x = -1, y = 2.

  • I use autograph to draw two lines on the same set of axes. To check the solution

image116.png

The solution of this 2×2 system of linear equations is unique.

  1. image124.pngimage124.png

image133.pngimage133.png

image145.pngimage145.png

image157.pngimage157.png

Substituting image159.pngimage159.png into equation (1): image160.pngimage160.png

image161.png

The solution is x = -1, y = 2.

2.image162.pngimage162.png

image163.pngimage163.png

(image164.pngimage164.png

image157.pngimage157.png

Substituting image159.pngimage159.png into equation (1): image165.pngimage165.png

image166.png The solution is x = -1, y = 2.

3.image167.pngimage167.png

image169.pngimage169.png

image170.pngimage170.png

image157.pngimage157.png

Substituting image159.pngimage159.png into equation (2): image171.pngimage171.png

image172.png

The solution is x = -1, y = 2.

4.image173.pngimage173.png

(2) ×image174.pngimage174.png: image175.pngimage175.png

(3) – (1):  image176.pngimage176.png

image08.pngimage08.png

Substituting image08.pngimage08.png into equation (1): image178.pngimage178.png

image179.png

...read more.

Middle

Now look at 3×3 system,

1. image21.pngimage21.png

By using GDC,

image22.pngimage23.pngimage24.png

From this, it is clearly shows there are infinite solutions. And the solution of this system isimage25.pngimage25.png , image26.pngimage26.png andimage28.pngimage28.png

Using 3-D sketch of autograph, we can see that three planes intercept each other at one line

image29.png

Therefore, image30.pngimage30.png.

2. image31.pngimage31.png

image32.pngimage33.pngimage34.png

It shows the solution of this system isimage25.pngimage25.png , image26.pngimage26.png andimage28.pngimage28.png.

3. image35.pngimage35.png

image36.pngimage23.pngimage24.png

Again it shows the solution of this system isimage25.pngimage25.png , image26.pngimage26.png andimage28.pngimage28.png.

4. image38.pngimage38.png

image39.pngimage33.pngimage34.png

Again it shows the solution of this system isimage25.pngimage25.png , image26.pngimage26.png andimage28.pngimage28.png.

5. image40.pngimage40.png

image42.pngimage23.pngimage43.png

From the above solution, my conjecture is: for any 3×3 system of linear equations which have the constants in the order of arithmetic sequences. They must have a infinite solution of image44.pngimage44.png , image45.pngimage45.png andimage46.pngimage46.png.      And x + y + z = 1

Therefore, I came out with a general system:

image47.png

image48.pngimage48.pngimage49.pngimage49.png

image50.pngimage50.png

image51.pngimage51.png

image52.pngimage52.png

image55.pngimage55.png

image56.pngimage56.png

Substituting image56.pngimage56.png into equation (1):

image57.pngimage57.png

image58.pngimage58.png

image59.pngimage59.png

Let image60.pngimage60.png

The solution of this system isimage25.pngimage25.png , image26.pngimage26.png andimage28.pngimage28.png

To test the validity of the conjecture,

image62.pngimage62.png

image63.pngimage64.pngimage65.png

There is limitation to this system: The ratio between the constants can’t be the same for the three of the equations.

Now look at 4×4 system,

1. image66.pngimage66.png

image67.pngimage68.png

The solution of this system isimage69.pngimage69.png , image70.pngimage70.png  andimage71.pngimage71.png.

2. image72.pngimage72.png

image73.pngimage68.png

Again it shows the solution of this system tobeimage69.pngimage69.png , image70.pngimage70.png  andimage71.pngimage71.png.

...read more.

Conclusion

image132.pngimage132.pngand common ratio of the second equation is image149.pngimage149.png.

image154.png

image155.png

image156.png

I used autograph, I observed that there are infinite solutions when two lines overlapped.

There is another limitation to this system: The common ratio of the two equations can’t be the same.

Conclusion

Part A

For any 2×2 system of linear equations which have the constants in the order of arithmetic sequences. They must have a unique solution of x = -1, y = 2. However the limitation to this system is the ratio between the constants can’t be the same for both of the equations.

For any n×n system (n>2) of linear equations which have the constants in the order of arithmetic sequences. They must have infinite solutions. However the limitation to this system is the ratio between the constants can’t be the same for the three of the equations. However the limitation to this system is the ratio between the constants can’t be the same for the all of the equations.

Part B

The 2×2 system of linear equations which have the constants in the order of geometric sequences. They must have a unique value. However, the limitations would be: The common ratio of the two equations can’t be the same or equal to 0.

...read more.

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