- Level: International Baccalaureate
- Subject: Maths
- Word count: 1460
Math IA patterns within systems of linear equations
Extracts from this document...
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:
Middle
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.
Part B
This is a new 2 x 2 system of linear equations:
Conclusion
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:
This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.
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