Patterns Within Systems Of Linear Equations

Patterns Within Systems Of Linear EquationsThe purpose of this report is to investigate dodges of analogue pars where the organizations ceaselesss have mathematical praxiss.The introductory system of rules to be considered is a 2 x 2 system of analog of equationsIn the first equation, the constants atomic number 18 1, 2, and 3 in that order. It is observed that each constant is increased by 1 from the previous constant. Thus, the constants cast off up an arithmetical grade whereby the first term ( U1 ) = 1, and the difference between each term ( d ) = 1. Hence, the world-wide impressula is Un = U1 + (n-1)(1) where n represents the nth term.U2 = U1 + (n-1)(d) U3 = U1 + (n-1)(d)2 = 1 + (2-1)(1) 3 = 1 + (3-1)(1)2 = 2 3 = 3In the indorse equation, the constants argon 2, -1, and -4 in that order. It is observed that each constant is increased by -3 from the previous constant. Thus, the constants from this equation also make up an arithmetic sequence whereby U1 = 2 and d = -3. Hence the general formula is Un = U1 + (n-1)(-3).U2 = U1 + (n-1)(d) U3 = U1 + (n-1)(d)-1 = 2 + (2-1)(-3) -4 = 2 + (3-1)(-3)-1 = -1 -4 = -4To further investigate the significance of these arithmetic sequences, the equations will be solved by electric switch and displayed graphically.x + 2y = 3 2x y = -4x = 3- 2y y = 2x + 4x = 3 2(2) y = 2(3 2y) +4x = -1 5y = 6 + 4y = 2On the graph, twain lines meet at a green point (-1,2) where x = -1 and y = 2.The two linear equations have a solution of x = -1 and y = 2, proven analytically and graphically.However, this pattern may be only specific to this 2 x 2 system of linear equations. Therefore, another(prenominal) 2 x 2 system of linear equations following the same pattern of having constants forming arithmetic sequences will be examined as well.Another 2 x 2 system of linear equations to be considered isThe constants of these equations ar 3, 6, and 9, and 4, 2, and 0 with a difference of 1 and -2 respectively.The equations were wher efore re-written asAnd plotted on a graph.The usual point of both equations is (-1,2), with x being -1 and y being 2. Therefore the common point has been proven both analytically and graphically to be (-1,2).Another example isThe constants of these equations are -3, 1, and 5, and -2, -6, and -10 with a difference of 4 and -4 respectively.The equations were then re-written asAnd plotted on a graph.The common point is (-1,2). Thus it is both proven analytically and graphically that the common point is (-1,2).Another example isThe constants of these equations are 3, 2, and 1, and 2, 7, and 12 with a difference of -1 and 5 respectively.The equations were then re-written asAnd plotted on a graph.The common point is (-1,2). Thus it is both proven analytically and graphically that the common point is (-1,2).Another example isThe constants of these equations are 5, 12, and 19, and 1, -5, and -11 with a difference of 7 and -6 respectively.The equations were then re-written asAnd plotted on a graph.The common point is (-1,2). Thus it is both proven analytically and graphically that the common point is (-1,2).From the examples of 22 systems of linear equations, a conjecture that could be derived isThe solution for any 22 system of linear equations with constants that form an arithmetic sequence is eer x=-1 and y=2.The general formula of such equations could be written asWhereby represents the first term for the first equation and represents the first term for the second equation with a common difference of and respectively.The equations are then solved simultaneouslyTherefore, it is proved that the solution for a 22 system of linear equations with constants that form an arithmetic sequence is always x = -1 and y = 2.However, the possibility of a 33 system exhibiting the same patterns as the previous 22 systems examined has not been discussed. Hence, this investigation will extend to 33 systems as well.Here is an 33 systemThe for the first equation is 3 and the is (5-3) = 2.The for the first equation is 1 and the is (-4-1)=-5.The for the first equation is 4 and the is (7-4)=3.Gaussian excretion method will be used.Change R3 into 4R2-R3Change R2 into 3R2-R1Change R3 into 23R2-17R3The third row/R3 has all 0 which means that in that respect is no one singular solution entirely infinite solutions. Therefore, in R2We will let where k is a parameterTo find other solutions, will be substituted in the other equationThe solutions to this 33 system of linear equations with the pattern of constants making up an arithmetic sequence are , , and where is a parameter.Here is another 33 systemThe for the first equation is 2 and the is (3-2)= 1.The for the first equation is 5 and the is (5-3)=-2.The for the first equation is -3 and the is (4-(-3))=7.The equations were prescribe into matrix form and row reduction was done on the computer graphic Design Calculator.The third row is all 0. This indicates that there is no unique solution, but infinite solutions in stead.Assuming that whereby is a parameter,The solutions to this 33 system of linear equations with the pattern of constants making up an arithmetic sequence are , , and where is a parameter.AnotherThe for the first equation is 4 and the is (-2-4)= -6.The for the first equation is 1 and the is (5-1)=-4.The for the first equation is 2 and the is (7-2)=5.The equations were put into matrix form and row reduction was done on the Graphic Design Calculator.The third row is all 0. This indicates that there is no unique solution, but infinite solutions instead.Assuming that whereby is a parameter,The solutions to this 33 system of linear equations with the pattern of constants making up an arithmetic sequence are , , and where is a parameter.Here is another 33 systemThe for the first equation is 4 and the is (-4-4)= -8.The for the first equation is 2 and the is (-1-2)=-3.The for the first equation is 6 and the is (14-6)=8.The equations were put into matrix form and row reduction was done on the Graphic Design Calculator.The third row is all 0. This indicates that there is no unique solution, but infinite solutions instead.Assuming that whereby is a parameter,The solutions to this 33 system of linear equations with the pattern of constants making up an arithmetic sequence are , , and where is a parameter.The for the first equation is 7 and the is (20-7)= 13.The for the first equation is 20 and the is (3-20)=-17.The for the first equation is 6 and the is (-5-6)= -11.The equations were put into matrix form and row reduction was done on the Graphic Design Calculator.The third row is all 0. This indicates that there is no unique solution, but infinite solutions instead.Assuming that whereby is a parameter,The solutions to this 33 system of linear equations with the pattern of constants making up an arithmetic sequence are , , and where is a parameter.From these examples, a conjecture can be made. A 33 system of equations that have constants that form an arithmetic sequence will have infinite solutions that will be in the form of , , and where is a parameter.This is proven by the general formula existence solved by using Gaussian elimination ruleChange R3 into R3-R2Change R2 into R2-R1Change R3 intoChange R2 intoChange R3 into R3-R2R3 has only zeroes/0. This means that there is no unique solution but infinite solutions instead.Assume whereby is a parameter,Through substitution,The solutions for this 33 system are , , and , proving the conjecture true.Other than systems of linear equations that contain arithmetic sequences, other types will be investigated.Lets consider this 2 x 2 systemIn the first equation, the constants 1, 2, and 4 make up a geometricalal sequence whereby the first term (U1) is 1 and each consecutive term is multiplied by a common ratio (r) which in this case is 2. U2 U3In the second equation, the constants 5, -1, and make up a geometric sequence whereby U1 = 5 and r = . U2 = U3The equations can be rewritten in the form of asFor the first equation, and .For the second equation, and..The relationship between and appears to be that one is the negative reciprocal of the other. In any case, more examples of similar linear equations will be needed to exhaustively investigate the patterns.The equations will be solved by substitutionAnother exampleIn the first equation, the constants 3, 12, and 48 make up a geometric sequence whereby the first term (U1) is 3 and each consecutive term is multiplied by r which in this case is 4. U2 U3In the second equation, the constants 3, -1, and make up a geometric sequence whereby U1 = 3 and r = . U2 = U3The equations can be rewritten in the form of asFor the first equation, and .For the second equation, and..The equations will be solved using substitutionAnother exampleIn the first equation, the constants 7, 42, and 252 make up a geometric sequence whereby the first term (U1) is 7 and each consecutive term is multiplied by r which in this case is 6. U2 U3In the second equa tion, the constants 2, -1, and make up a geometric sequence whereby U1 = 2 and r = . U2 = U3The equations can be rewritten in the form of asFor the first equation, and .For the second equation, and..The equations will be solved by using substitutionFrom observing all three systems, it is found that the relationship between and appears to be that one is the negative reciprocal of the other. But it can also be say that .The general formula of such equations could be written asWhereby represents the first term for the first equation and represents the first term for the second equation with a common ratio of and respectively.The equations are then solved simultaneouslySo is the result of one ratio subtracted from the other.is the product of the common ratios from both linear equations.