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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 equations

• Let the equations be ax + by=c,

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

For the second equation ( ): 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:

Substituting x= -1 into equation (4):

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

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

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

Substituting  into equation (1):

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

2.

(

Substituting  into equation (1):

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

3.

Substituting  into equation (2):

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

4.

(2) ×:

(3) – (1):

Substituting  into equation (1):

Middle

Now look at 3×3 system,

1.

By using GDC,

From this, it is clearly shows there are infinite solutions. And the solution of this system is ,  and

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

Therefore, .

2.

It shows the solution of this system is ,  and.

3.

Again it shows the solution of this system is ,  and.

4.

Again it shows the solution of this system is ,  and.

5.

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  ,  and.      And x + y + z = 1

Therefore, I came out with a general system:

Substituting  into equation (1):

Let

The solution of this system is ,  and

To test the validity of the conjecture,

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.

The solution of this system is ,   and.

2.

Again it shows the solution of this system tobe ,   and.

Conclusion

and common ratio of the second equation is .

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.

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

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