# A new formulae method for solving the simultaneous equations

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INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME 6, ISSUE 03, MARCH 2017ISSN 2277-8616A New Formulae Method For Solving The Simultaneous…
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INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME 6, ISSUE 03, MARCH 2017ISSN 2277-8616A New Formulae Method For Solving The Simultaneous Equations Avinash A. Musale, Eknath S. Ugale ABSTRACT: Actually there are too many Methods which are used for solving the various types of equations by various methods. The equations may be like linear, quadratic, simultaneous, radial, and exponential as well as the equations with the number of variables. And for the Simultaneous Equation the methods are like Addition, Substitution, Elimination as well as Graphical Method also. We know there are too many methods like Gauss Elimination, Gauss Seidal, and Jacobi used for solving the Simultaneous Equations. But here we are going to introduce a new Method for solving the Simultaneous Equations as well as the comparison between the Methods above told and the new one. The paper will show you a new easy method for solving the Simultaneous Equation and the difference between the methods that are being used in the colleges which are Gauss Elimination, Gauss Seidal etc. Keywords: Simultaneous, Equations, Formulae, Method, Gauss, Elimination, Derivation ————————————————————1. INTRODUCTION: The paper represents a method for solving the simultaneous equations in a different way. We use the substitution, addition, matrices Methods for solving the simultaneous equations but in this paper the method is related to directly the formulas for solving the simultaneous equations.2. METHOD (FORMULAE): For Two Unknowns – X & Y a11X+a12Y=b1; a21X+a22Y=b2; Solution: u= [(a11*b2 - a21*b1)] v= [(a11*a22 - a12*a21)] Y= X= For Three Unknowns – X, Y & Z a11X+a12Y+a13Z=b1; a21X+a22Y+a23Z=b2; a31X+a32Y+a33Z=b3; Solution: r= [(a11*b3) - (a31*b1)]; s= [(a11*b2) - (a21*b1)]; t= [(a11.a22) - (a12*a21)]; u= [(a11*a33) - (a13*a31)]; v= [(a11*a23) - (a13*a21)]; w=[(a11*a32) - (a12*a31)];+Z=* Y=*+ +X=*For Four Unknowns – X, Y, Z & W a11X+a12Y+a13Z+a14W=b1; a21X+a22Y+a23Z+a24W=b2; a31X+a32Y+a33Z+a34W=b3; a41X+a42Y+a43Z+a44W=b4; Solution: k=[(a11*b4) - (a41*b1)]; l= [(a11*b3) - (a31*b1)]; m= [(a11*b2) - (a21*b1)]; n= [(a11.a22) - (a12*a21)]; o= [(a11*a33) - (a13*a31)]; p=[(a11*a44) - (a14*a41)]; q= [(a11*a23) - (a13*a21)]; r=[(a11*a24) - (a21*a14)]; s=[(a11*a32) - (a12*a31)]; t=[(a11*a34) - (a31*a14)]; u=[(a11*a42) - (a41*a12)]; v=[(a11*a43) - (a41*a13)]; g=(on-sq) h=(vn-uq) i=(ln-sm) j=(tn-sr) e=(kn-um) f=(pn-ur) +W=* Z=*  Avinash A. Musale is currently pursuing bachelor degree in Mechanical Engineering in MIT AOE Pune in Savitribai Phule Pune University, India, Mobile No. +918484972706. E-mail: musaleavinash24@gmail.com  Mr. Eknath S. Ugale is an Assistant Professor in the Department of Mechanical Engineering in MIT AOE Pune, India, E-mail: esugale@mech.mitaoe.ac.inY=* X=*+ + +77 IJSTR©2017 www.ijstr.orgINTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME 6, ISSUE 03, MARCH 20173. DERIVATION:ISSN 2277-8616or from equation (b) We getFor Two Unknowns â&#x20AC;&#x201C; X & Y+X=*a11X+a12Y=b1; a21X+a22Y=b2;orX=*+If u=[(a11*b2 - a21*b1)] v=[(a11*a22 - a12*a21)]Using Gauss Elimination Method Convert the given equations in matrix AX=B form đ?&#x2018;&#x17D;11 đ?&#x2018;&#x17D;12 đ?&#x2018;&#x2039; đ?&#x2018;?1 * +** +=* + đ?&#x2018;&#x17D;21 đ?&#x2018;&#x17D;22 đ?&#x2018;&#x152; đ?&#x2018;?2Y=Make the Matrix A as Upper Triangular Matrix X=Make row 1 as following Instruction R1 = R1/a11 đ?&#x2018;&#x17D;12â&#x20AC;˛+ * *đ?&#x2018;&#x2039;+ = *đ?&#x2018;?1â&#x20AC;˛+ * 1 đ?&#x2018;&#x152; đ?&#x2018;&#x17D;21 đ?&#x2018;&#x17D;22 đ?&#x2018;?2 a12â&#x20AC;&#x2122;=a12/a11 b1â&#x20AC;&#x2122;=b1/a11For Three Unknowns â&#x20AC;&#x201C; X, Y & Z a11X+a12Y+a13Z=b1; a21X+a22Y+a23Z=b2; a31X+a32Y+a33Z=b3;â&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Ś.(1) â&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Ś.....(2)By Gauss Elimination Method Convert the given equations in matrix AX=B formMake row 2 as following Instruction R2 = R2 - a21*R1 đ?&#x2018;&#x2039; *1 đ?&#x2018;&#x17D;12â&#x20AC;˛+ * * + = *đ?&#x2018;?1â&#x20AC;˛+ đ?&#x2018;&#x152; 0 đ?&#x2018;&#x17D;22â&#x20AC;˛ đ?&#x2018;?2â&#x20AC;˛ a22â&#x20AC;&#x2122;=a22 - a21*a12â&#x20AC;&#x2122; b2â&#x20AC;&#x2122;=b2 â&#x20AC;&#x201C; a21*b1â&#x20AC;&#x2122;đ?&#x2018;&#x17D;11 [đ?&#x2018;&#x17D;21 đ?&#x2018;&#x17D;31â&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Ś (3) â&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Ś(4)Make the Matrix A as Upper Triangular Matrix therefore Make row 1 as following Instruction R1 = R1/a11 đ?&#x2018;&#x2039; 1 đ?&#x2018;&#x17D;12â&#x20AC;˛ đ?&#x2018;&#x17D;13â&#x20AC;˛ đ?&#x2018;?1â&#x20AC;˛ [đ?&#x2018;&#x17D;21 đ?&#x2018;&#x17D;22 đ?&#x2018;&#x17D;23 ] * [đ?&#x2018;&#x152; ] = [ đ?&#x2018;?2 ] đ?&#x2018;? đ?&#x2018;&#x17D;31 đ?&#x2018;&#x17D;32 đ?&#x2018;&#x17D;33 đ?&#x2018;?3Make row 2 as following Instruction R2 = R2/a22â&#x20AC;&#x2122; đ?&#x2018;&#x2039; *1 đ?&#x2018;&#x17D;12â&#x20AC;˛+ * * + = * đ?&#x2018;?1â&#x20AC;˛ + đ?&#x2018;&#x152; 0 1 đ?&#x2018;?2â&#x20AC;˛â&#x20AC;˛ b2â&#x20AC;&#x2122;â&#x20AC;&#x2122;=b2â&#x20AC;&#x2122;/a22â&#x20AC;&#x2122;â&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Ś(5)We Getâ&#x20AC;Ś. đ?&#x2018;&#x2039; *1 đ?&#x2018;&#x17D;12â&#x20AC;˛+ * * + = * đ?&#x2018;?1â&#x20AC;˛ + đ?&#x2018;&#x152; 0 1 đ?&#x2018;?2â&#x20AC;˛â&#x20AC;˛Y=**+ +*+a22â&#x20AC;&#x2122;= a22 - a21*a12â&#x20AC;&#x2122; a23â&#x20AC;&#x2122;= a23 - a21*a13â&#x20AC;&#x2122; b2â&#x20AC;&#x2122; = b2 â&#x20AC;&#x201C; a21 * b1â&#x20AC;&#x2122;â&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Ś.(4) â&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Ś.(5) â&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Ś.(6)a32â&#x20AC;&#x2122; = a32 â&#x20AC;&#x201C; a31 * a12â&#x20AC;&#x2122; a33â&#x20AC;&#x2122; = a33 â&#x20AC;&#x201C; a31 * a13â&#x20AC;&#x2122; b3â&#x20AC;&#x2122; = b3 â&#x20AC;&#x201C; a31â&#x20AC;&#x2122; * b1â&#x20AC;&#x2122;â&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Ś.(7) â&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Ś.(8) â&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Ś..(9)1 [0 0a11 gets cancelledâ&#x20AC;Ś..and from given equationâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Ś (1) â&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Ś(2) â&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Ś.(3)Make row 2 as following Instruction R2 = R2/a22+Y=*a12â&#x20AC;&#x2122;= a12/a11 a13â&#x20AC;&#x2122;=a13/a11 b1â&#x20AC;&#x2122;=b1/a11Make row 2 and row 3 as following Instruction R2 = R2 â&#x20AC;&#x201C; a21* R1 R3 = R3-a31 *R1 đ?&#x2018;&#x2039; 1 đ?&#x2018;&#x17D;12â&#x20AC;˛ đ?&#x2018;&#x17D;13â&#x20AC;˛ đ?&#x2018;?1â&#x20AC;˛ [0 đ?&#x2018;&#x17D;22â&#x20AC;˛ đ?&#x2018;&#x17D;23â&#x20AC;˛] * [đ?&#x2018;&#x152; ] = [đ?&#x2018;?2â&#x20AC;˛] đ?&#x2018;? 0 đ?&#x2018;&#x17D;32â&#x20AC;˛ đ?&#x2018;&#x17D;33â&#x20AC;˛ đ?&#x2018;?3â&#x20AC;˛From the above solved Matrix By the Backward Substitution we will get Here Y=b2â&#x20AC;&#x2122;â&#x20AC;&#x2122; â&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Ś.(a) X+a12â&#x20AC;&#x2122;Y=b1â&#x20AC;&#x2122; â&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Ś(b) And from equation (a) Y=b2â&#x20AC;&#x2122;â&#x20AC;&#x2122; From eq(5) Y=* + From equation (3),(4) Y=* + From equation (1),(2) Y=*đ?&#x2018;&#x17D;12 đ?&#x2018;&#x17D;13 đ?&#x2018;&#x2039; đ?&#x2018;?1 đ?&#x2018;&#x17D;22 đ?&#x2018;&#x17D;23] * [đ?&#x2018;&#x152; ] = [đ?&#x2018;?2] đ?&#x2018;&#x17D;32 đ?&#x2018;&#x17D;33 đ?&#x2018;? đ?&#x2018;?3+đ?&#x2018;&#x2039; đ?&#x2018;&#x17D;12â&#x20AC;˛ đ?&#x2018;&#x17D;13â&#x20AC;˛ đ?&#x2018;?1â&#x20AC;˛ 1 đ?&#x2018;&#x17D;23â&#x20AC;˛â&#x20AC;˛] * [đ?&#x2018;&#x152; ] = [đ?&#x2018;?2â&#x20AC;˛â&#x20AC;˛] đ?&#x2018;? đ?&#x2018;&#x17D;32â&#x20AC;˛ đ?&#x2018;&#x17D;33â&#x20AC;˛ đ?&#x2018;?3â&#x20AC;˛a23â&#x20AC;&#x2122;â&#x20AC;&#x2122; = a23â&#x20AC;&#x2122;/a22â&#x20AC;&#x2122; b2â&#x20AC;&#x2122;â&#x20AC;&#x2122; =b2â&#x20AC;&#x2122;/a22â&#x20AC;&#x2122;â&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Ś..(10) â&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Ś..(11)Make row 3 as following Instruction 78 IJSTRÂŠ2017 www.ijstr.orgINTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME 6, ISSUE 03, MARCH 2017R3 = R3 â&#x20AC;&#x201C; a32â&#x20AC;&#x2122; * R2 đ?&#x2018;&#x2039; 1 đ?&#x2018;&#x17D;12â&#x20AC;˛ đ?&#x2018;&#x17D;13â&#x20AC;˛ đ?&#x2018;?1â&#x20AC;˛ [0 1 đ?&#x2018;&#x17D;23â&#x20AC;˛â&#x20AC;˛] * [đ?&#x2018;&#x152; ] = [đ?&#x2018;?2â&#x20AC;˛â&#x20AC;˛] đ?&#x2018;? 0 0 đ?&#x2018;&#x17D;33â&#x20AC;˛â&#x20AC;˛ đ?&#x2018;?3â&#x20AC;˛â&#x20AC;˛*(Z=* Z= *[â&#x20AC;Śâ&#x20AC;Ś..(b)*+â&#x20AC;Ś..fromeq(7,8,9,10,11)*()+ *((()) [(( ))+ [ ()()]) ] )Z= *()+ *(([))(())]] *] [+] *[+] [] [] []]+]+ ++] []] [] [] []+]+))+ [From equation (b) Y+a23â&#x20AC;&#x2122;â&#x20AC;&#x2122;Z=b2â&#x20AC;&#x2122;â&#x20AC;&#x2122;))] (())(())))+ [Y=[(b2â&#x20AC;&#x2122;â&#x20AC;&#x2122;)-(a23â&#x20AC;&#x2122;â&#x20AC;&#x2122;*Z)]] (())]Y= *+-*+from eq(10,11)â&#x20AC;Ś..From eq(1,2,3) ((()*(())+ *(())+ [Z= ()*()+ *((*] (())(())(())))+ [[]([)*(+)+ *(((Y=[((Z=+Y=* ())+ *(()]as well as a11 gets cancelled )*(((Y=*+ gets cancelled(+Y=*))))+ [((())(())(())](Y=[ *))â&#x20AC;Ś..from eq(4,5,6) ))))) (((Y=[]( ((())]))+ [)) (((())) [(()) ((( (] â&#x20AC;Ś..from eq(1,2,3)))) (()])) ))])]+ will get cancelledâ&#x20AC;Ś 79IJSTRÂŠ2017 www.ijstr.org+]If r= [(a11*b3) - (a31*b1)]; s= [(a11*b2) - (a21*b1)]; t= [(a11.a22) - (a12*a21)]; u= [(a11*a33) - (a13*a31)]; v= [(a11*a23) - (a13*a21)]; w=[(a11*a32) - (a12*a31)]; Z=*(())+ *((((]â&#x20AC;Śâ&#x20AC;Ś. From eq (1,2,3,4,5,6) *((] [Z= [ *[â&#x20AC;Śâ&#x20AC;Ś.from eq(14)))+ (*))đ?&#x2018;&#x17D;11 đ?&#x2018;&#x17D;22 â&#x2C6;&#x2019; đ?&#x2018;&#x17D;21 đ?&#x2018;&#x17D;12 gets cancelledâ&#x20AC;Ś Z= [ ] [ * [[ ] [ *â&#x20AC;Śâ&#x20AC;Ś.fromeq(12)&(13)([] (())+ [*â&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Ś..(14)Z=**()+ *(([â&#x20AC;Śâ&#x20AC;Ś..(a)Z=[))+ [[â&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Ś(12) â&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Ś(13)Z = b3â&#x20AC;&#x2122;â&#x20AC;&#x2122;â&#x20AC;&#x2122; Y+a23â&#x20AC;&#x2122;â&#x20AC;&#x2122;Z=b2â&#x20AC;&#x2122;â&#x20AC;&#x2122; From equation (a) We get Z = b3â&#x20AC;&#x2122;â&#x20AC;&#x2122;â&#x20AC;&#x2122; Z=* +=))+ *((Z=Make row 3 as following Instruction R3 = R3/a33â&#x20AC;&#x2122;â&#x20AC;&#x2122; đ?&#x2018;&#x2039; 1 đ?&#x2018;&#x17D;12â&#x20AC;˛ đ?&#x2018;&#x17D;13â&#x20AC;˛ đ?&#x2018;?1â&#x20AC;˛ [0 1 đ?&#x2018;&#x17D;23â&#x20AC;˛â&#x20AC;˛] * [đ?&#x2018;&#x152; ] = [ đ?&#x2018;?2â&#x20AC;˛â&#x20AC;˛ ] đ?&#x2018;? 0 0 1 đ?&#x2018;?3â&#x20AC;˛â&#x20AC;˛â&#x20AC;˛ b3â&#x20AC;&#x2122;â&#x20AC;&#x2122;â&#x20AC;&#x2122; = b3â&#x20AC;&#x2122;â&#x20AC;&#x2122;/a33â&#x20AC;&#x2122;â&#x20AC;&#x2122; From the above Matrix, By the Backward Substitution we getZ))((*((a33â&#x20AC;&#x2122;â&#x20AC;&#x2122; = a33â&#x20AC;&#x2122; - a32â&#x20AC;&#x2122; * a23â&#x20AC;&#x2122;â&#x20AC;&#x2122; b3â&#x20AC;&#x2122;â&#x20AC;&#x2122; = b3â&#x20AC;&#x2122; â&#x20AC;&#x201C; a32â&#x20AC;&#x2122; * b2â&#x20AC;&#x2122;â&#x20AC;&#x2122;+ISSN 2277-8616+]INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME 6, ISSUE 03, MARCH 2017() [((Y=[)) ((( () (()) ))(Y=*)])ISSN 2277-8616And finally ]X=3 Hence we can conclude+X=3 Y=-1 Z=-2s= [(a11*b2) - (a21*b1)]; t= [(a11.a22) - (a12*a21)]; v= [(a11*a23) - (a13*a21)];B. New Formula Method x-3y+z=4 2x-8y+8z=-2 -6x+3y-15z=9Y=* + And from given equations we get X=* + And here Simultaneously we can derive the formulas for the four unknowns in the same way above derived. As well as we can also derive the formulas for the Simultaneous equations with the number of derivatives.Compare it with a11X+a12Y+a13Z=b1; a21X+a22Y+a23Z=b2; a31X+a32Y+a33Z=b3;4. COMPARISIONThen here for three unknowns Substitute the values in the given formulae Hence we will getA. Gauss Elimination Method x-3y+z=4 2x-8y+8z=-2 -6x+3y-15z=9r= [(a11*b3) - (a31*b1)]; r= 33Convert into AX=B Matrix form Make the Matrix A as Upper Triangular Matrix 1 â&#x2C6;&#x2019;3 1 đ?&#x2018;&#x2039; 4 [ 2 â&#x2C6;&#x2019;8 8 ] * [đ?&#x2018;&#x152; ] = [â&#x2C6;&#x2019;2] 9 â&#x2C6;&#x2019;6 3 â&#x2C6;&#x2019;15 đ?&#x2018;?t= [(a11.a22) - (a12*a21)]; t= -2Make row 2 and row 3 as following Instruction R2 = R2 â&#x20AC;&#x201C; 2* R1 R3 = R3-(-6) *R1 1 [0 0â&#x2C6;&#x2019;3 â&#x2C6;&#x2019;2 â&#x2C6;&#x2019;151 đ?&#x2018;&#x2039; 4 6 ] * [đ?&#x2018;&#x152; ] = [â&#x2C6;&#x2019;10] 33 â&#x2C6;&#x2019;9 đ?&#x2018;?s= [(a11*b2) - (a21*b1)]; s= -10u= [(a11*a33) - (a13*a31)]; u= -9 v= [(a11*a23) - (a13*a21)]; v=6 w=[(a11*a32) - (a12*a31)]; w=-15Make row 2as following Instruction R2 = R2/ (-2)Z=*1 [0 0Z= -2â&#x2C6;&#x2019;3 1 â&#x2C6;&#x2019;151 đ?&#x2018;&#x2039; 4 â&#x2C6;&#x2019;3] * [đ?&#x2018;&#x152; ] = [ 5 ] 33 â&#x2C6;&#x2019;9 đ?&#x2018;?Make row 3 as following Instruction R3 = R3 â&#x20AC;&#x201C; (-15) * R2 1 [0 0â&#x2C6;&#x2019;3 1 01 đ?&#x2018;&#x2039; 4 â&#x2C6;&#x2019;3 ] * [đ?&#x2018;&#x152; ] = [ 5 ] 108 â&#x2C6;&#x2019;54 đ?&#x2018;?And then by Backward Substitution Method We will get -54*Z=108 Z=-2+=*Y=* +=* Y= -1 X=* X=3[] [ []] [[+= *]]+++Hence X=3, Y= -1, Z= -2 Hence here is the solution by the formulae methodY-3*Z=5 Y=-1 80 IJSTRÂŠ2017 www.ijstr.orgINTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME 6, ISSUE 03, MARCH 2017ISSN 2277-86165. CONCLUSION: Here we can conclude that there is no confusion in solving the simultaneous equation by the Formulae Method. As well as it is easier and takes less time than the other methods like Gauss Elimination Method, Gauss Seidal Method, etc. And in the Engineering Calculators like CASIO fx-991ES PLUS there is no solution for the simultaneous equation with four unknowns so we can use these formulas for solving the Simultaneous equations easily as well as fast.6. REFERENCES: Steven C. Chapra, Raymond P. Canale, Numerical Method for Engineers. 4/e, Tata McGraw Hill Editions, 2002, ISBN 0-07-047430Dr. B. S. Garewal, Numerical Method in Engineering and Science, 7/e, Khanna Publishers, ISBN81-74009-205-6Steven c. Chapra, Applied Methods in MATLAB for Engineers and Scientist, 2/e, Tata McGrawHill Publishing Co.Ltd, 2008, ISBN007-064853-0M. K. Jain, S R K Iyengar, R K Jain, Numerical Method for Scientific and Engineering Computations, 5/e New Age International Publisher ISBN 13 978-81-224-2001-2008Gary D. Knott, "Gaussian Elimination and LUDecomposition", www.civilized.com, Feb 28, 2012Joseph F. Grcar, ―Mathematicians of Gaussian Elimination‖, Notices of the AMS, June 2011Nondeterministic Procedure of Solving Simultaneous Equations, N. Shobha Rani, Research scholar, Maharaja Research Foundation, MIT, Mysore, International Journal of Engineering and Science Issn: 2278-4721, Vol. 2, Issue 3 (February 2013)81 IJSTR©2017 www.ijstr.org

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