• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month

# 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

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.

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.

## Found what you're looking for?

• Start learning 29% faster today
• 150,000+ documents available
• Just £6.99 a month

Not the one? Search for your essay title...
• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month

# Related International Baccalaureate Maths essays

1. ## Extended Essay- Math

B ï¿½F<ÝÛ­-ï¿½ï¿½Õ1/4W1/2Èº1ï¿½ï¿½ï¿½c?2ï¿½ï¿½ Qï¿½b9 ï¿½Ó¸i21Ú`(c)ï¿½?ï¿½ï¿½ï¿½[K2ï¿½jEQ-ï¿½"UKï¿½TTMï¿½ï¿½ï¿½ï¿½ï¿½0fï¿½ï¿½y...ï¿½:ï¿½cï¿½ï¿½W~&ï¿½ ï¿½_ï¿½@;~TKï¿½(c)ï¿½bï¿½ï¿½@ku_I]@ï¿½ï¿½ï¿½+CÛ¤ mï¿½ ï¿½ï¿½ï¿½V1/2a-yCgï¿½ï¿½sJ ï¿½mï¿½Þ1Iï¿½`ï¿½0 ï¿½#ï¿½×+ï¿½nï¿½..Cï¿½#wï¿½l#|ï¿½ ï¿½Gï¿½3å`ï¿½(r)ï¿½ ï¿½"o-dJï¿½&FÚ¯ï¿½-'ï¿½*ï¿½Xï¿½"ï¿½"E?(tm)ï¿½/ï¿½6rqoï¿½Ç¯Cï¿½;ï¿½ï¿½}ï¿½Qï¿½ï¿½M(r)ï¿½ï¿½]ï¿½wï¿½v+ï¿½zWï¿½ "ï¿½ï¿½ï¿½'ï¿½ï¿½ï¿½ï¿½jï¿½"ï¿½ï¿½-vfqf-ï¿½ï¿½¦ï¿½ ï¿½É²"Oeï¿½ ï¿½z-Sï¿½Ë´1/2ï¿½è±-ï¿½ï¿½\ï¿½ï¿½?sï¿½ï¿½iï¿½1/2_Pï¿½Wï¿½0r'ï¿½ ï¿½ï¿½Tï¿½ 8ï¿½ï¿½9"Wï¿½?l>ï¿½ÞY_Ð(ï¿½ï¿½ï¿½:"0ï¿½ï¿½ddMï¿½TÊsaï¿½ï¿½sï¿½aï¿½ï¿½|*:3>ï¿½ï¿½ï¿½BWbFï¿½ï¿½ï¿½sRvï¿½ï¿½ï¿½Ø·ï¿½ï¿½ï¿½ 5 Nk oï¿½.ï¿½(c)1ï¿½ï¿½(c)Lï¿½:"h wï¿½ï¿½ï¿½ï¿½D"@)ï¿½ï¿½1/2ï¿½ï¿½(tm)ï¿½ï¿½^fï¿½ï¿½|ï¿½,ï¿½+Tï¿½ï¿½Hï¿½ï¿½""f'ê£ï¿½Hï¿½*@ï¿½"Gçeï¿½ï¿½3ï¿½ï¿½[uï¿½søï¿½"#Sï¿½:(tm)Eï¿½zï¿½)w"ï¿½ï¿½ dï¿½-"a~1/2E2Jrï¿½Tï¿½&j3ï¿½bï¿½Dsï¿½}ï¿½ï¿½ï¿½-ï¿½\$"ï¿½mï¿½T:3ï¿½[email protected]ï¿½Bï¿½ï¿½ï¿½Eï¿½ ?ï¿½\ 1}zï¿½Ð0Eï¿½ï¿½ï¿½8ï¿½v (r)ï¿½[email protected] 2lYï¿½A! ^ï¿½7ï¿½2ï¿½3/4ï¿½ï¿½?uï¿½0ï¿½ï¿½(tm) ^Cbï¿½^ï¿½rï¿½Bï¿½;ï¿½ï¿½ï¿½ Lï¿½@&8+ï¿½"'ï¿½ ï¿½ï¿½"[email protected]ï¿½ï¿½ï¿½ï¿½faï¿½0ï¿½ ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½eï¿½RGï¿½lnfA2 ï¿½,ï¿½wKï¿½ 1/2"tï¿½&Pï¿½`ï¿½%\$?ï¿½ï¿½ï¿½Mï¿½ï¿½C=ï¿½a- ï¿½"?ï¿½ï¿½ï¿½"ï¿½ï¿½ï¿½ï¿½+ï¿½ï¿½ï¿½...rï¿½Yï¿½ï¿½ï¿½ï¿½4.ï¿½_ï¿½ï¿½ï¿½\ï¿½ï¿½\$ï¿½ï¿½//[ï¿½Vï¿½aï¿½,ï¿½ï¿½Jï¿½(c)i ddï¿½ qï¿½;qpyï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½yVï¿½ï¿½ï¿½f\ï¿½K<[ï¿½ï¿½<ï¿½ï¿½ï¿½; )ï¿½\ï¿½ Oï¿½Kï¿½ï¿½ï¿½:ï¿½`Ü¶ï¿½ ï¿½Zï¿½mï¿½ï¿½ï¿½ï¿½ï¿½%!B<dR}ï¿½%ï¿½.Fï¿½ï¿½Yìxï¿½ï¿½2ï¿½ï¿½Ø¶ï¿½ï¿½ï¿½Iï¿½1/2ï¿½ï¿½FCï¿½ï¿½:[ï¿½ï¿½,ï¿½ï¿½ï¿½ ^"Hï¿½ï¿½ï¿½ï¿½pï¿½Pï¿½ï¿½ï¿½ï¿½x4ï¿½ï¿½ s)ï¿½bCï¿½nhï¿½ï¿½@U\$"ï¿½ï¿½ï¿½q ï¿½<'ï¿½=ï¿½pï¿½.1ï¿½2Ú´ï¿½ï¿½Ê±ï¿½ï¿½ï¿½aËb1/4ï¿½ï¿½ï¿½Fï¿½ï¿½ï¿½7ï¿½ï¿½ï¿½Éµi2j`ï¿½6...ï¿½ï¿½B ï¿½Tï¿½MWï¿½ ï¿½bï¿½ï¿½ ï¿½ï¿½(tm)

2. ## Math Studies I.A

one may say they chose it in random there is always some form of bias. Furthermore, while choosing randomness of selection can result in a sample that doesn't reflect the makeup of population or unlucky error. Stratified sampling is not used because it is difficult to categorise countries based on any characteristics.

1. ## Math IA - Logan's Logo

The maximum value is (1.4, 0.9) and the minimum is(-2.0, -3.5). We can now substitute the x-values of these two points into the period equation to find b: TO FIND c: As previously stated, changing the variable c will affect the horizontal shift of the sine curve, so thatare translations to the right, whileare translations to the left.

2. ## A logistic model

Graphical plot of the fish population of the hydroelectric project on an interval of 30 years using the logistic function model {19}. The model considers an annual harvest of 8x103 fish. A stable population (4x104) is reached. c. Consider an initial population u1=3.5x104 fish.

1. ## Math IA - Matrix Binomials

of a (a= -2): For An when a=10: When n=1, 2, 3, 4, ... (integer powers increase), then the corresponding elements of each matrix are: 1, 20, 400, 8000, ... These terms represent the pattern between the scalar values multiplied to A=aX where a=10 and hence A= to achieve an end product of An.

2. ## Math Studies - IA

Nevertheless, two sets of data rarely have the same sample size. Final scores from the 96 rounds of golf and 3 Ryder Cups can be found directly from the tournaments' websites. To collect data from the majors, the players who participated in the Ryder Cup will again be divided into two teams: US and Europe.

1. ## Creating a logistic model

Looking at the graph for r=2.9, we see that these fluctuations in fact, get larger, until it settles down at a certain limit. The same can be said for the graph for r=3.2, where the population of fish oscillates in a pattern over time.

2. ## MATH IA- Filling up the petrol tank ARWA and BAO

For Arwa’s Vehicle E2 =p2(r +d× day/w)/f l =p2(20km+d/10)/(18km) ∴p2 =(E2×18km)/(20km+d/10) y=p2 and x=d ∴The lines are of the form y=(E2×18)/(20+x/10), where E2 is a constant. ∴The line is an inverse function. We observe that as d is increasing, p2 is decreasing and hence p2 is indirectly proportional to d when E2 is constant. • Over 160,000 pieces
of student written work
• Annotated by
experienced teachers
• Ideas and feedback to 