# 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.

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):

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.

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 ...