math hl portfolio (aps and gps)

ACADEMIA BRITANICA CUSCATLECA

MATH HL PORTFOLIO

PATTERNS WITHIN SYSTEMS OF LINEAR EQUATIONS

Benjamin Valiente

8/29/2010

This document contains a mathematical investigation referring to systems of linear equations, divided into two parts A and B, one about constant forming an AP and the other about constants forming a GP.


ContentsIntroduction:..............................................................................................................3

Part A (Constants forming APs):................................................................................4

What is meant by “constants forming APs”?..........................................................4

Examining the Constants:.......................................................................................4

Looking for a pattern:.............................................................................................4

Solving other systems:............................................................................................5

Conclusion and Proof for a 2 x 2 system:...............................................................7

Investigating 3 x 3 systems:...................................................................................8

Conjecture and Prove of a 3 x 3 system:..............................................................11

Part B (Constants forming GPs):..............................................................................14

Examining the given 2 x 2 system:.......................................................................14

Investing the relationship between the constants:...............................................14

Looking for patterns:............................................................................................15

Solving a general 2 x 2 system that follow the pattern found:.............................16

Proving the graphical pattern:..............................................................................17

Finding a General Solution for a 2 x 2 system:.....................................................18

General Conclusion:.................................................................................................21

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Introduction:As IB mathematic HL we should have encountered “arithmetic progressions” (APs) and “geometric progressions” (GPs), as well as systems of linear equations, for instance simultaneous equations. When we form APs or GPs with the constants of these systems of linear equations, the equations should form patterns. This investigation is designed to show that they do form a pattern, and if they do to find a mathematical proof why these patterns are formed by the equations.

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Part A (Constants forming APs):

What is meant by “constants forming APs”?The constants are the numbers which aren’t variables in an equation.

For instance in the equation before a, b and c are constants in the equation. So they form APs when a, b and c have a “common difference” (d).

So the constants, a, b and c form an “arithmetic progression”.

Examining the Constants:We are given two simultaneous equations, if we inspect the constants we get this:

So after inspecting both equations we can observe that the constants of both equations form APs.

Looking for a pattern:I will solve the system of simultaneous equations above, to try to find any pattern that they may form, due to the fact that their constants form an AP.

If I use the method of elimination the first equation has to be multiplied by 2 so that both equations can be subtracted.

Now we can use the elimination method.

So after we substitute y into one of the two equations to get the value of x.

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So the point where both of these equations cross has the coordinate (-1, 2)

This graph is a graphical representation of the equation solved afore. In the graph it is clearly shown that the coordinate found algebraically is the same as shown on the graph, proving that the lines cross at the point (-1, 2).

The observation that can be made about the solution of the simultaneous equations is the fact that the coordinates are equal to the 2nd term of the APs, in other words the constant of y. The x-coordinate, -1, is equal to the y constant on the

equation . And the y-

coordinate, 2, is equal to the constant

of y on the equation .

Solving other systems:Now I will try other systems which are similar to see if there is any pattern formed by the solution of these equations.

Example 1:

Using the same method of elimination this equations can be solved.

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Now we substitute y into one of the equations to get the value of x.

Intersection point coordinates is (-1, 2)

Example 2:

To use elimination the first equation has to be multiplied by 3 so that the second equation can be subtracted from it.

Now we can use elimination with both equations.

Again we substitute y into one of the equations

Intersection point coordinates is (-1, 2)

Graphing the equations:

This is a graphical representation of the 6 afore solved.

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As the algebraic solution and the graphical solution show, all the lines formed by

the equations depicted above intercept at the same point, (-1, 2).

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Conclusion and Proof for a 2 x 2 system:All the equations that I have solved before intercept at point (-1, 2); and all the equation that I have solved above have constants which form an AP. So this leads to form a conclusion: that all the equations which have constants that form APs will all intercept at point (-1, 2).\

But what if it is coincidence of just those six equations? Well, it can be proved algebraically that all the equations which have constants that form APs will all intercept at point (-1, 2).

Let’s assume that a is the first term of the AP formed by one equation’s constant, and that it has a common difference “d”, so it the equation can be written as:

And let’s assume that for another equation which is simultaneous to the one before, b is the first term of the AP formed by its constants, and it also has a common difference of “e”, so it is written as:

So now these two equations can be solved simultaneously, by elimination method.

But we can’t subtract them that, so I will multiply the first equation by b and the second equation by a.

Now they can be subtracted:

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Now that we know that y=2, we substitute y into one of the equations.

Now that algebraically I proved that x=-1 and that y=2, I have proved that all it works for all numbers, because a, b, d and e can be substituted by any numbers and the answer will always be x=-1 and y=2. So any 2 x 2 system of linear equations which have constants that form APs will always intercept at the point, (-1, 2).

Investigating 3 x 3 systems:Now, what will happen if instead of a 2 x 2 system I had a 3 x 3 system of equations? Will it form a common line, there will be no pattern? To answer these questions I will create three 3 x3 system of equations, and whose constants form an AP.

The equations are:

System 1:

System 2:

System 3:

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If we graph the three systems we get this graph:

We can see from both of the views of this graph that the planes all intercept on a common line. So as the 2 x 2 system has a common point, the 3 x 3 system has a common line.

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But what is the equation of the line? If we see from the graph that the three systems intercept on the same line, then if I solve just one of them I will get the equation of the common line.

I will solve system 3, I will use matrices and row reduction to solve this system.

So the matrix will be:

Now with row reduction I will solve this system of simultaneous equations.

So after row reduction we see that the equation has a common line and not a common point, something we expected to see as the graph showed to us. So now we know that:

Assume that , therefore

Or

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So now we can we can substitute y and z into the equation which has the three variables.

Now we have three equations for in terms of the three different variables, x, y

and z.

So if they all equal it means that they all equal each other so the following

equation is true.

The equation above is the equation of the line where the three systems that I graphed before intercept, denoting that it is the common line for the three 3 x 3 systems conveyed before.

Conjecture and Prove of a 3 x 3 system:Observing the pattern made by the three 3 x 3 systems solved before and at the result gotten from the 2 x 2 systems, I can predict that any plane whose constants form an AP will always pass through the line

which means that any 3 x 3 system of equation whose constants form an AP will all intercept on the same, which is mentioned before.

What about if it was just a coincidence for those three systems? How do we know that it will work for all equations which comply with the condition of constants forming an AP? As the 2 x 2 system of equations the 3 x 3 system of equations can

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also be proved algebraically. However this time we will use row reduction instead of elimination to solve the system of equations.

Let’s assume that the first term of the AP formed by the constants of an equation with three variables is a, and that it has a common difference d.

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Now for the second equation let’s assume that the first term formed by the constants is b, and that it has a common difference of e.

Finally, for the last equation let’s assume that the first term formed by the constants is c, and that it has a common difference of f.

With the three simultaneous equations, now I can form a matrix.

Now with this matrix I can do row reduction to solve the system of equations.

As expected the three equations have a common line because we can observe on R3 000=0, so now we know that:

Assume that , so:

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And

Now we have a value for y and z we can substitute them into the other equation:

So now we have three equations for in terms of x, y and z:

So if they all equal they all equal each other, so we can say,

The afore graphed planes all intercepted on the line that I just got by using algebra to solve the simultaneous equation. This proofs that my conjecture was right because a, b, c, d, e and f could be replaced by any number and it will always give the same line, so all 3 x 3 systems which have constants that form an AP will always intercept on the same line;

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Part B (Constants forming GPs):

Examining the given 2 x 2 system:We are given two equations

To look for a pattern I will isolate the constants of both of these equations

We get:

And

If we observe them, they form a “Geometric Progression”.

What is a GP?

Any sequence/progression that has a common ratio “r”, this means that each term is multiplied by this value.

For instance in the first equation r=2, because:

And for the second , because

So for these equations the constants form a GP compared to the ones on part A which formed an AP.

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Investing the relationship between the constants:To find any relationship the equation has to be written in a different format;

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So I will write the two equations mentioned before in the new format:

And

Let’s take the second equation because it is easier to see any existent pattern, so

So if we examine the values of a and b we can see a clear relationship:

Or

Looking for patterns:To look for patterns I will graph several line, this will help me search for any graphical patterns that could exist when the constants for a GP.

I will use 10 different lines, which are:

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If we observe the graph lines are all over the Cartesian plane except for a kind of parabola that is formed on the left hand side of the

graph, where . It seems to be

a type the area where there are no lines has a contour which forms a parabola which corresponds to a

.

Solving a general 2 x 2 system that follow the pattern found:The pattern found in the graph has a mathematical explanation, and the contour of the area with no lines has an equation. To find this equation I will generate a general equation which complies with the rule that the constants form a GP.

Let us assume that for the first term of the GP formed by the constants of a linear equation is “a”, and that the common ratio is “r”. So,

The a cancels out, so we get

We see that we have a quadratic equation in terms of r, therefore we state that:

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For r needs to be a real number because it is the common ratio, and you can’t have an imaginary common ratio, so for r to exist the following condition need to be true:

If

Then

In this case

So will be true

Now we solve it,

So for r to be real and for the GP to exist, as well as for the 2 x 2

systems to exist.

Proving the graphical pattern:

If we plot we get a

graph that is depicted. It seems similar to the area that where no lines cross on the graph done before. What would happen if I superimpose this graph over the one I had before?

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This is what I get when I superimpose the two graphs. This shows that the area contour of the area which doesn’t have lines going through it is exactly the

same as the line . This is

completely logical because within that are r doesn’t exist, it is imaginary, which means that the lines also don’t exist so if they don’t exist they don’t appear on the Cartesian plane leaving that space.

This proofs the graphical pattern observed on the graph to the right.

Finding a General Solution for a 2 x 2 system:Let us write two general equations that comply with the constants forming a GP.

a is the first term and r will be the common ratio for the first general equation. For the second equation b is going to be the first term and q is going to be the common ratio. Therefore:

If we solve this pair of simultaneous equation we will get a general solution for the equations. So by elimination we solve the system:

The a and the b get eliminated as they are constants throughout the equation.

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Now that we have an equation for y, as a solution for the general 2 x 2 system, we use substitution to find the solution for x.

We now have the general solution:

𝑦=𝑟+𝑞𝑥=−𝑟𝑞To prove that the solution is right I will solve a 2 x 2 system of simultaneous equation which follow the pattern and see if the interception point given by the above solution equals the graphical solution of the system.The equations are:

Now we find the values of r in the first equation and q in the second equation. To do this we divide the third term by the second:

With these two values we can now use the general solution to find the interception point.

Therefore the interception point is (-0.4, 2.2).To see if the general solution works with the system of equations portrayed above, I am going to solve the system through elimination method.25 | P a g e


To solve the equation we need to have one of the coefficients to be equal so I multiply the first equation by 25.

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So:

Now we substitute the value of y into one of the equations to get the value of x.

It can be observed that the values achieved through both methods coincide, so we see that the general solution for a 2 x 2 system of linear equations, which have constants that form a GP, will be:

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General Conclusion:After this investigation is clearer than in mathematics everything is related, there are no coincidences. For instance, having a pattern in the constants of different systems of equations, 2 x 2 or 3 x 3, or even an AP as the pattern or a GP, they all have a resulting pattern; and that these patterns have a mathematical proof, which makes them real and logical. After doing this investigation I will feel comfortable predicting that if I investigated constants forming an exponential progression or a 3 x 3 system which constants formed a GP, they will both form a resulting pattern with a clear, logical mathematical proof.

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math hl portfolio (aps and gps)

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