Patterns Withing Systems of Linear Equations - Math Problem Example

? School School Crest Mathematics HL Portfolio Type I PATTERNS WITHIN SYSTEMS OF LINEAR EQUATIONS Candi Candi Number Examination Session Contents Introduction 2 Part A. System of linear equations formed with arithmetic progressions. 5 Arithmetic progressions 5 Constants in a system of linear equations 5 Searching for a pattern 6 Pattern repeated in other systems 8 Graphing equations with arithmetic progress coefficients 10 Generalize result for systems 11 Extension to systems of linear equations 13 Part B. System of linear equations formed with geometric progressions. 19 Geometric progressions 19 Examining the given systems 20 Relationship between constant within geometric progressions 20 Family of linear equations to discovering a new pattern 21 General solution for system that follows the pattern found 23 Graphical prove of the pattern found 25 Conclusion. 26 Introduction Algebra is one of the fascinating fields of mathematics, because algebra allows the finding of unknown numbers from information given. In algebra, letters are used in place of numbers that are not known. These letters are then manipulated in accordance with certain rules until an answer appears. The usual letter for the unknown number is. A real problem can be written as: This is called an equation because there is a sign. In order to find the value of the unknown number, algebra’s rules can do whatever it likes to this equation as long as it does the same to both sides of the equation. So far it has had equation with a single unknown number. What if it has two unknown numbers? In fact, an equation with two unknown has an infinite numbers of pairs of answer. To fix a single pair of number as the answer, it needs another equation. A pair of equation, each with two unknown numbers is called simultaneous equations. They can be solved together to give the values for the unknowns that satisfy both equations simultaneously. This paper contains a mathematical research about systems of linear equation when their coefficients obey arithmetic or geometric progressions. An arithmetic progression is a sequence of numbers where each number is a certain among larger than the previous one. The numbers in the sequence are said to increase by a common difference, d. For example: is an arithmetic progression where the. The term of this sequence is. On the other hand, a geometric progression is a sequence where each number is times larger than the previous one. is known as the common ratio of the progression. The term of a geometric progression, where is the first term and is the common ratio, is: . For example, in the following geometric progression, the first term is , and the common ratio is : the term is therefore. The purpose of this portfolio is to show how with the aid of technology using appropriate computer software likes Autograph and Maxima packages (see Figure 1) is quick and easy to get graphical representations of algebraic equations. Thus, how in many situations, the graphs offers much more insight into the problem than does the algebra. Part A will consider the patterns within systems of linear equations:, where and are in arithmetic progression. While, in Part B the same coefficients obey geometric progression. Part A. System of linear equations formed with arithmetic progressions. Arithmetic progressions In algebra, letters are used in place of numbers that are not known. The usual letter for the unknown numbers are or . . The numbers are constants in an equation, for example: For instance in the above equation, and are known as constants in the equation. It says that the constant and form a arithmetic progression if they have a common difference, such as: Constants in a system of linear equations Given the system of linear equations. The coefficients are detected as follow: Examining the first equation, it sees a pattern in the constants of the equation. i.e. is the constant preceding the variable , and precede and the equation equals 3. The constant have a common difference of between them: Going forward from this, it can say that the constants in the first equation belong to an arithmetic series with a common difference of. This means they follow the general formula: = number of the term (i.e. first, second, third, etc) = in the common difference in the series. Looking at the second equation, it also observes that the constants are consecutive of and arithmetic series, this time with common difference of. Searching for a pattern In order to find any pattern that the system of linear equations may form, due to the fact that their constants form an arithmetic progression. It process as follow: To solve the equations simultaneously, it can be used the method of elimination, thus the first equation has to be multiplied by to eliminate variable. Then, both equations can be added: Elimination method yields: or Putting in the solution obtained for into the original equation to find: Finally, The point where both of these equations cross has the coordinate. Figure 2: It shows the graphical solution of the systems of linear equations given above. The solutions is the point Using technology, with the aid of Autograph the figure 2 is the graphical representation of the equation solved above. In the graph it is clearly shown that the coordinate found algebraically is the same as shown on the graph (black point), proving that the lines cross at the point. It has to remember that for two-variable system linear of equations, there are three possible types of solutions. The first case is presented by two distinct non-parallel lines that cross at exactly one point. This is called an “independent” system of linear equations and the solution is always some -point. The second case is presented by two distinct lines that are parallel. Since parallel lines never cross, then there can be no intersection; that is, for a system of linear equations that graphs as parallel lines, there can be no solution. This is called an “inconsistent” system of linear equations, and it has no solution. The third case is considered when only one line appears in the graph. Actually, it is the same line drawn twice. These two lines, really begin the same line, “intersect” at every point along their length. This is called a “dependent” system of linear equation, and the “solution” is the whole line. This shows that a system of linear equation may have one solution, no solution or infinite solutions. It will never have a system of linear equation with two or three solutions; it will always be one, none, or infinitely-many. Pattern repeated in other systems It 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 system can be solved as: Now substitute into the second of the equations to get the value of, thus: Intersection point coordinate is . Example 2 To use elimination method the second equation has to be multiplied by to eliminate: Substituting into second equation: Intersection point coordinate is . Example 3 To use elimination method the second equation has to be multiplied by to eliminate: Substituting into first equation: Intersection point coordinate is . Graphing equations with arithmetic progress coefficients In order show all steps above in a graph, it has been used autograph to display all system linear equations. As the algebraic solution predicts, the graphical solutions show, all the lines formed by the equations depicted above intersect at the common solution, . Figure 3: It shows graphical representation of the above system of linear equations. All systems have the same and unique solution. The black point is the common solution. Generalize result for systems It seems that any system of linear equations where the constants follow an arithmetic progression will give the answer for and . This leads to form a conclusion: that all the systems of linear equations which have constants that form an arithmetic progression will all intercept at point. Is it coincidence of just those 8 equations analyzed above?. This means that it can generate a general formula for equations similar to these: Let the first term of the first equation be and let if have a common difference of . Let the first term of the second equation be and have a common difference of . Therefore the first equation will look like: The second equation will look like: Now these two equations can be solved simultaneously, by elimination method: multiply the first equation by to eliminate multiply the second equation by to eliminate subtract the second equation from the first one. After little algebra, Finally, Putting the value for in the first equation: Finally, Now that algebraically it proved that and , thus it has proved that all it works for all numbers, because and can be substituted by any numbers and the answer will always be the same: . The results suggest that for any two linear equations written in the form: which have constants and that follows an arithmetic progression will always have solution at the point: . Extension to systems of linear equations What will happen if instead of a systems of linear equations it had a systems of linear equations?. Will there be no pattern? Will it form a common point?. In order to investigate this situation, it has been extended the system up to systems for instance using the following examples: Example 1: Solve this system of equation using matrices: To solve for and we have to find the inverse of the matrix A, and multiply it with A: We used Maxima software to calculate the inverse of A: Figure 4: It uses Maxima to calculate the inverse of the matrix A. However, matrix A is singular or determinant equal to zero. This is singular matrix, this means the determinant = 0. Therefore there are either many solutions for or there is no solution at all: Example 2: We used Maxima software to calculate the inverse of the matrix, see Figure 5: It shows that the analyzed matrix is singular or determinant equal to zero. It seems that all systems that exhibit an arithmetic progression will give a determinant of zero, i.e. form a singular matrix. Let the first term of the first equation be and the common difference. Let the first term of the second equation be and the common difference. Let the first term of the third equation be and the common difference. The equations will look like: Arranging the equation into a matrix: Finding the inverse of matrix A is given by: Thus, first find the determinant of matrix A: This shows that all matrices with arithmetic progression will have a determinant of 0. This means there is either a contradiction in the equations i.e. no definite answer, or there are many answers, i.e. they intersect in a line. In order to find out if there are no solutions we have to see if there is a contradiction in the equations: First the three are grouped in groups of two as shown: Group I. Multiply the first equation by to eliminate: Multiply the second equation by. Apply elimination method. Group II. Multiply the second equation by to eliminate: Multiply the third equation by . Apply elimination method. First two equations (group I): Subtract the second from the first. With the first two equations we find if Last two equations (group II) Subtract the second from the first. Two solutions show us that there is no contradiction in the three equations; this means that the equation has many solutions, i.e. they intersect not a point, but they intersect to form a line. If we plot the equations: We get the following graph using Autograph computer package: Figure 6: It shows a system of linear equations. The system of linear equations has common point, while the system of linear equations has common line. Note that the intersection form the line: . This shows us that the three equations join to make a line. Following the equation of the line: when when We have two points and , we can use the two points to find the direction vector of the line : Direction vector: Therefore we can say that for a system that exhibit an arithmetic sequence, the three plains will meet to form a line that has the direction vector: or a multiple of it. The point on the line has to satisfy the equation: where is the first term of the arithmetic series and is the common difference of the series. Part B. System of linear equations formed with geometric progressions. Geometric progressions A geometric progression is a sequence where each number is times larger than the previous one. is known as the common ratio of the progression. A geometric progression has the following properties: Each term follow the general formula , where is the first term of the series, is the term number (i.e. first, second, third, etc) and is the common ratio. Thus, we studied system with geometric progression on its coefficients. Examining the given systems Considering the system of linear equations is given by They exhibit a geometric progression in the constants of the equations. In the first equation the constant are: The second constant is two times the first, and the third is two times the second. The equation exhibits a geometric series with a common ratio of . For second equation the constants are: The common ratio is , i.e. if we take the first constant and multiply it by the common ratio we get the second constant and so on. Relationship between constant within geometric progressions To find any relationship, the system has to be written in a different format of the form. Thus, after writing the two equations mentioned before in the new format: We see that the constants have a common feature; is the negative inverse of , and is the negative inverse of , i.e. they are negative inverses of each other. This means if we multiply the constant by the results is -1. Family of linear equations to discovering a new pattern In order to discover a pattern in this section, we will graph several lines, this will help to search for any graphical patterns that could exist when the constant of a system are related with geometric progressions. The following list of line equations have been used to draw Figure 7. All these equations obey geometric progression in their coefficients: Figure 7: It shows several system equations that obey geometric progression in their coefficients. The line graph cover the Cartesian plane except for a kind of parabola region 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, for some value of k. Notice that there is a region on the Figure 7 which has no intersections; this is a pattern on the graph. What regions is this, and from where does it come?. It is clear that the lines graph cover the Cartesian plane except for a kind of parabola region that is formed on the left hand side of the graph, where . The pattern found on the Figure 7 has a mathematical explanation, and the contour of the area with no lines has an equation. To find this equation, we generate a general equation which complies with the rule that the constants form a geometric progression. Assumes that for the first term of the geometric progression formed by the constant of a linear equation is and that the common ratio is . Thus, The constant cancels out, so it yields There is a quadratic equation in terms of , therefore the equation to be solve is: Furthermore, needs to be real number because is the common ratio, then for to exist the following condition needs to be true. The discriminate of the above equation has to satisfy: Now it solves: Thus, for to be real and for the geometric progression to exist, as well as for the system to exist, the following condition on -point must be satisfied: or Thus, the null region appears to be a contour given by a parabola of the form. General solution for system that follows the pattern found In order to generalize the above result, it needs to find solutions to the different lines and it can be calculated using the general formula, thus: Geometric series with first term and common ration of Geometric series with first term and common ration of Multiplied the first equation by to eliminate Multiplied the second equation by to eliminate After some algebra, When we plug in the value for in our original equation: Therefore the intersection point is given at To prove the formula, we found the point of intersection for the systems of linear equations analytically and graphically: common ration of common ration of Intersection at: 0.4 Additionally, we used autograph to illustrate the solution as shown in the Figure 8. Figure 8: It shows the intersection at the point of the system equation: and . Notice that the same formula can be used to calculate the different points of intersection for the different graphs. Graphical prove of the pattern found The Figure 9 shows that the area contour which doesn’t have lines going through it is exactly the same as the line . This is completely logical because within that are 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 empty. Figure 9: It shows the contour of the area which doesn’t have lines going through it is exactly the same as the line. Notice that the solutions of the line graph shown in Figure 9 has the form: and . That is . To see this, combine all terms so that , and notice that the left side of the equation is a perfect square which is always greater than or equal to zero. The boundary for the null region seen in Figure 9 is . Conclusion. After relating two separates topics in mathematics such as system of linear equation with numerical progressions, we get amazing results. For example, having a pattern in the constants of systems of equations, such as or or even an arithmetic progression as the pattern or a geometric progression, they all have a resulting pattern; and that these patterns have a mathematical proof, which makes them real and practical. References Gareth W., 2011 “Linear Algebra with Applications” Ed. Jones and Bartlett Publishers. Read More