Simultaneous equations 8 Introduction to advanced methods

Transcription

1 1 Simultaneous equations 8 Introduction to advanced methods J A Rossiter For a neat organisation of all videos and resources Slides by Anthony Rossiter

2 Introduction 2 The earlier videos gave an introduction to simultaneous equations and simple solution methods used in schools. The following videos look at methods which are more useful for larger problems, many of which are not amenable to paper and pen solutions. We start with simple problems to illustrate concepts and limitations, and then finally move to computer efficient methods such as Gaussian elimination. Slides by Anthony Rossiter

3 Solving for a single unknown Let s assume that we have only one equation. Which possibilities do we have? a 0 then x = b a a 1 b ax = b Unique solution for x 3 a = 0, b = 0 Infinitely many solutions a = 0, b 0 then ax = b 0 = b ax=b has no solution for x Slides by Anthony Rossiter

4 Key point 4 Not all simultaneous equations have: Unique solutions. Or indeed any solution at all. Before attempting to solve simultaneous equations, it is worth asking whether a solution exists. Sometimes the test for this is easy and other times less so. Slides by Anthony Rossiter

5 Solving two equations in 2 unknowns where a, b, c, d, e, f are given numbers Multiply equation (i) by d, equation (ii) by b and subtract Multiply equation (i) by c, equation (ii) by a and subtract What if ad-bc=0?

6 Illustration of no solution 6 Solve 2x+y=3, 4x+2y=8. Subtracting twice the 1 st equation from the 2 nd equation gives 0 = 2 which is inconsistent. NO SOLUTION. In this case it is clear that ad-bc = 0 Parallel lines Slides by Anthony Rossiter

7 Solving two equations in 2 unknowns What variation in parameters a, b, c, d, e, f admit: 1. a unique solution. 2. no solution or an infinite number of solutions. In general it is quite complicated to quantify the 2 nd of these, especially for simultaneous equations with multiple unknowns. Here, it is sufficient to note that a unique solution exists, if and only if the left hand side coefficients of the variables satisfy: ad-bc 0 We come back to this shortly.

8 8 USING MATRICES WITH SIMULTANEOUS EQUATIONS Slides by Anthony Rossiter

9 Expressing a system of equations using matrices One important applications of matrices is for the solution of linear simultaneous equations. Let s start with a simple example : A X = B A X B X = A -1 B

10 Put following into matrix-vector format z y x y z y z x y x y x

11 Importance of the determinant After representing linear simultaneous equations in matrix format, the determinant has a core role. Unique solution if ad-bc 0 A X B A X = B A = a c b d ad bc = A Unique solution iff det(a) 0

12 Explanation Assume the linear simultaneous equations have been represented as: 12 AX = B Then the solution is given as X = A -1 B However, a matrix inverse A -1 is only defined if the determinant is non-zero. If the determinant is zero, there is either no solution or an infinite number of solutions. Slides by Anthony Rossiter

13 Solving of system of equations B = 4 1 X=A -1 B

14 Solve the following: 4x-3y=2, 5x+2y=-1

15 Represent using matrices A 1 = ; X = A 1 B Again, it can be shown that there is a unique solution if and only if det(a) 0.

16 Summary This brief resource introduced the concepts of simultaneous equations having either a unique solution, no solution, or an infinite number of solutions. It has shown that simultaneous equations can be conveniently represented using a matrix/vector format of the coefficients. It has been indicated that a unique solution can only exist if the matrix of coefficients has a non-zero determinant; the solution can be defined with the matrix inverse. The next video on how to compute matrix inverses will clarify this. 16 Slides by Anthony Rossiter

17 Anthony Rossiter Department of Automatic Control and Systems Engineering University of Sheffield For a neat organisation of all videos and resources /indexwebbook.html 2018 University of Sheffield This work is licensed under the Creative Commons Attribution 2.0 UK: England & Wales Licence. To view a copy of this licence, visit or send a letter to: Creative Commons, 171 Second Street, Suite 300, San Francisco, California 94105, USA. It should be noted that some of the materials contained within this resource are subject to third party rights and any copyright notices must remain with these materials in the event of reuse or repurposing. If there are third party images within the resource please do not remove or alter any of the copyright notices or website details shown below the image. (Please list details of the third party rights contained within this work. If you include your institutions logo on the cover please include reference to the fact that it is a trade mark and all copyright in that image is reserved.)