• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month
Page
  1. 1
    1
  2. 2
    2
  3. 3
    3
  4. 4
    4
  5. 5
    5
  6. 6
    6
  7. 7
    7
  8. 8
    8
  9. 9
    9
  10. 10
    10
  11. 11
    11
  12. 12
    12
  13. 13
    13
  14. 14
    14
  15. 15
    15
  16. 16
    16
  17. 17
    17
  18. 18
    18
  19. 19
    19
  20. 20
    20
  21. 21
    21
  22. 22
    22
  23. 23
    23
  24. 24
    24
  25. 25
    25
  26. 26
    26

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.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related International Baccalaureate Maths essays

  1. In this task, we are going to show how any two vectors are at ...

    figure 3 The slope of these two point is same as the slope of the equation . (Refer to figure 2) We assume that the vector is so = td {distance = time � speed} Now by drawing the diagram t ?

  2. Math IA Type 1 In this task I will investigate the patterns in the ...

    Therefore the conjecture needs to be refined to: Now I will look to find if changing the quadrant in which the vertex is placed, alters or refines my conjecture. To analyze this I will place the vertex in each of the quadrants.

  1. Parabola investigation. In this task, we will investigate the patterns in the intersections of ...

    Consider the parabola , the lines and. By using Graphmatica software, we can obtain the four intersections of parabola , the lines and. The x-values of these intersections from the left to the right on the x-axis: > x1�

  2. Moss's Egg. Task -1- Find the area of the shaded region inside the two ...

    A Moss's Egg = + 2 + 2 + 2 - 9 � 35.8 cm2 b) Determining the perimeter of Moss's Egg involves similar processes as that in finding its area, as this too, involves breaking up Moss's Egg into its smaller segments.

  1. Math IA patterns within systems of linear equations

    (first term: -2, common difference: -2) Multiplying the first equation by -2 gives: And subtracting the second equation from this equation gives: Therefore Substituting for y in the first equation gives: So The solution is again x=-1 and y=2. We can consider more equations, where the coefficients follow an AS: -> x=-1 -> y=2 When we

  2. Maths IA. In this task I am asked to investigate the positions of ...

    Side?s A to O and A to P? are both 1, because they are the radiuses of point A (midpoint of triangle C3), I am told that r=1(r being the radius). In the second part of this problem I have to find the distance from O to P?, considering that O to P = 3.

  1. Develop a mathematical model for the placement of line guides on Fishing Rods.

    Since the length of the rod is finite (230cm) then the number of guides is known to be finite. Domain = , where n is the finite value that represents the maximum number of guides that would fit on the rod. Dependent Variable: Let y represent the distance of each guide from the tip of the rod in centimetres.

  2. Infinite Summation- The Aim of this task is to investigate the sum of infinite ...

    Now a general sequence where x=1, will be considered. Taking all data collected into account a relation between a and ? has shown. So as the two sets of results show a=2 or a=3, the total of sum won?t exceed 2 or 3, when x=1. To prof this suggestion right, two more examples will be made, again when x=1

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work