Math IA patterns within systems of linear equations

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Let’s consider a second system, where the coefficients also follow an AS but with different first terms and different common differences:

                 (first term: 1, common difference: +2)

         (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 graph all equations, we see that all lines intersect at the point (-1;2) We can make the conjecture, that all 2 x 2 systems of linear equations where the coefficients follow an AS have the same solution of x=-1 and y=2.

To prove this conjecture we consider a general 2 x 2 system of linear equations:

         (first term: a, common difference: d)

         (first term: b, common difference: e)

In order to solve these simultaneous equations by elimination, we need to multiply the first equation by b and the second equation by a. This gives us:

Now we can subtract the second equation from the first which gives us:

Now we can substitute 2 for y in one of the equations. Doing this with the first equation gives:

The solution to the general 2 x 2 system of linear equations is x=-1 and y=2. Therefore we can say that this is always the solution to a 2 x 2 system  of linear equations, no matter what numbers we put in for a, b, d or e. We have proven our conjecture to be true.

We now extend our investigation to a 3 x 3 system. First we create three 3 x 3 systems of equations where the constants of every equation follow an AS:

System (1)

System (2)

System (3)

This is the graphical display of the equations:

We can solve the first system with a 3 x 3 matrix using technology:

Now we know that

Assume

 therefore

We also know that

As

 we can say

Now we have three equations for t which we can set equal.

This is the equation in which all planes of system (1) meet.

General conjecture

Because all of the panes graphed before intersect in the same line we can come up with the conjecture that all 3 x 3 systems of equations where the constants follow an AS intersect in the line

 .

In the general model of 3 x 3 system a, b and c are always the first terms and d, e and f the common differences.

This system we can solve by row reduction:

As R3 000=0 we can say that the planes meet in one line. If we assume

we know from R2:

From that we get

        and        

.

Now we can substitute for y and z:

Now we have three equations:

          and        

                and        

The all equal t so we can say:

This is the same equation as we got before and the algebraic proof that all the planes intersect in that line. No matter values a, b, c, d, e, or f have, any 3 x 3 system of equations where the constants follow an AS will intersect in the line

. We have proven our conjecture to be true.

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Part B

This is a new 2 x 2 system of linear equations:

If we look at the coefficients we get 1, 2, 4 for the first equation and 5, -1,

 for the second equation. Both form a geometric sequence (GS). A geometric sequence has common ratio (r) which is the ratio of two consecutive terms. The common ratio in the first equation is 2 and in the second equation it is

.

Rewriting the two equations so they fit the format

 results in:

We notice that in both equations the constants a and b are related in the formula

        or         

.

To get a better picture of the idea we will use 8 different equations where the coefficients all follow this rule:

 

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We can see that the lines are distributed everywhere except for an area which seems to be beneath the function

.

General solution to 2 x 2 system following a GS

Because we have assumed that the coefficients in the equations follow the pattern of a GS we can produce a general model:

where a is the first term and r the common ratio. The a cancels out so we are left with:

This is a quadratic equation in terms of r so we rearrange it to read:

Because r is a ratio it cannot be imaginary and must be real. Therefore we can say that the conjugate has to be positive. In the general formula

 the conjugate is

. In our case a=1, b=-y and c=-x. We can say that

.

When we solve this:

So for r to be a real number and a GS to exist,

.

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The graphical pattern

If we now superimpose the function

 (in red) onto the graph from before we can see that all lines lie outside the area between the red curves.

General solution to 2 x 2 system with constants that follow a GS

We can produce a general model for a 2 x 2 system of linear equations where the constants follow a GS if we say a is the first term and r the common ratio in the first equation. In the second equation b is the first term and q the common ratio.

This system of simultaneous equations we can solve by elimination. We can divide out a in the first equation and b in the second equation. So we subtract what we end up from the second equation from the first equation:

Now we can substitute for y in the first equation. This gives us:

Proof of the general solution

To prove this we consider the 2 x 2 system of equations

In this system r=2 and q=-1/2. Using the two formulas for x and y we received above we get:

         ->        

        ->        

If we now solve the system by elimination for x and y:

Substituting x = 1 into the first equation gives us:

We received the same solutions. This tells us our formulas for x and y in terms of r and q are correct. We can say that the solution to any 2 x 2 system of linear equations where the constants follow a GS will be: