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.

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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.

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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):

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

5.

   

 

 

               

Substituting  into equation (1):

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

From the five 2×2 system of linear equations I have investigated, all of them have a unique solution of x = -1, y = 2.

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Therefore, my conjecture is: 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.

And I came out with these general equations:

  

 

                 

                       

                           

Substituting  into equation (1):

This system has a unique solution of x ...

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