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Math IA patterns within systems of linear equations

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Introduction

Math HL Investigation – Maximilian Stumvoll

Math HL Investigation

Patterns within Systems of

Linear Equations

Maximilian Stumvoll

11/11/2012

LIS


Part A

We consider this 2 x 2 system of linear equations:

x+2y=3

2x-y=-4

Looking at the coefficients in the first equation (1, 2, 3) we notice a pattern. Adding 1 to the coefficient of x (1) gives the coefficient of y (2) and adding 1 once more gives the constant (3).A similar pattern exists in the second equation. Only, here we add -3 to the coefficients. We can say the coefficients follow an arithmetic sequence (AS). An arithmetic sequence has a common difference (d). This is the difference between two consecutive terms of the sequence. The first equation has a first term (a) of 1 and a common difference (d) of 1. The second equation has a first term (b) of 2 and a common difference (e) of -3.

In order to solve this system of equations with the method of solving simultaneous equations by elimination, we need to multiply the first equation by 2. Then we can subtract the second equation from the first equation:

As proof we can solve the simultaneous equations by the method of substitution. Rewriting the first equation as

 and substituting for x in the second equation gives:

.

This we can solve for y:

Substituting this y-value into the second equation gives:

...read more.

Middle

 we can say

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

image00.png

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 image00.png

 .

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        image00.png

.

Now we can substitute for y and z:

Now we have three equations:

          and        image00.png

                and        

The all equal t so we can say:

image00.png

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 image00.png

. We have proven our conjecture to be true.


Part B

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

image00.png

...read more.

Conclusion

image07.png

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.

image02.png

image02.png

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:

image02.png

image02.png

image03.png

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

image02.png

image03.png

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:

image00.png

         ->        

image00.png

        ->        image00.png

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

Substituting x = 1 into the first equation gives us:

image00.png

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:

...read more.

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

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