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 = 1, y = 2.
To test the validity of the conjecture,
1.
2.
3.
4.
5.
There is limitation to this system: The ratio between the constants can’t be the same for both of the equations. If the ratios are same, two lines are overlapped, and the solutions become infinite.
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 3D 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.
From the above solution, my conjecture is: for any 4×4 system of linear equations which have the constants in the order of arithmetic sequences. They must have a infinite solution of , and. And
Now look at 5×5 system,
1.
This shows the solution of this system is , and.
And
2.
Again this shows the solution of this system is , and.
From the above solution, my conjecture is: for any 5×5 system of linear equations which have the constants in the order of arithmetic sequences. They must have infinite solutions of , and.
And
Now look at the system of equations,
When n˃2, there are infinite solutions. The general solution is
...
And.
There is limitation to this system: The ratio between the constants can’t be the same for the all of the equations.
Part B
Consider the 2×2 system of linear equations,
I find that the constants follow an order of geometric sequences,
For the first equation: , the constants (1, 2, 4) starts with 1 and has a common ratio of 2.
For the second equation: , the constants (5, 1, ) starts with 5 and has a common ratio of .
Rewrite the two equations in the form of y = ax + b:
First equation:
Second equation:
The relationship between constants a and b is
By using autograph to draw a line which. And vary the constant b from
100 to100. So there are 201 lines on the same set of axes.
By zooming out the graph, we would be able to see,
It shows a Ushape region on the left of the graph which no line intersects.
The general 2×2 system which the constants of each equations is a geometric sequence is,
a is the common ratio of equation 1 and b is the common ratio of equation 2.
Substituting into equation (1):
There is a unique solution of this 2×2 system of equations. The general solution is and .
To test the validity of the solution,
1.
From this system, I observed the common ratio for the first equation is and common ratio for the second equation is .
The general solution state that and .
Therefore,
I used autograph to find the intersection point of the lines is at (0.5, 2.833).

For this system, the common ratio of the first equation is and common ratio of the second equation is .
I used autograph to find the intersection point of the lines is at (2, 3).

For this system the common ratio of the first equation is and common ratio of the second equation is .
There is limitation to this system: The common ratio of the two equations can’t be 0.

For this system, the common ratio of the first equation is 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.