Math IA patterns within systems of linear equations

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Math HL Investigation – Maximilian Stumvoll

Math HL Investigation

Patterns within Systems of

Linear Equations

Maximilian Stumvoll



Part A

We consider this 2 x 2 system of linear equations:



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:

Solving for x gives:

Therefore the only solution satisfying both equations is x=-1 and y=2. This means that the point (-1;2) is the only point that lies on both lines. It is the point of intersection.

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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:


Substituting for y in the first equation gives:


The solution is again x=-1 and y=2.

We can consider more equations, where the coefficients follow an AS:


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