where the a's are constants, the x's the variables and b's are again constants.

Transcription

1 Phys_108B_Linear_Eqn. Page 1 Systems of Simultaneous Linear Equations Simultaneous equations are those that are satisfied by the same sets of values for the variables - there can be lots of types of simultaneous equations but the type we are going to look at is linear equations of two or more variables a k1 x 1 + a k2 x 2 + a k3 x = b 1 a m1 x 1 + a m2 x 2 + a m3 x = b 2 where the a's are constants, the x's the variables and b's are again constants. Example: x + y = 3 x - 2 y = 4 By the way, it is very important that our equations be independent of each other; that is that one of them is not simply a rearrangement of another, the sum of others or another one multiplied by a constant These equations are independent x + y = 4 x - 2 y = 4 These are not independent of those above x = 4 - y (rearrangement of the first) 2x + 2y = 8 (first multiplied by 2) (to illustrate the sum of two equations not being independent we would have to have three variables) Simultaneous linear equations show up in many places - in about a week in Phys 220 lecture we are going to be solving complicated electrical circuits which have many independent equations; also in business applications they show up a lot, in maximizing profits from a fixed inventory for example (case such as 4 walls + 2 doors = 1 small house at 25,000; 6 walls + 4 doors = larger house at 40,000. You have 100 walls, 90 doors what is mix of small and large

2 Phys_108B_Linear_Eqn. Page 2 houses you make to get greatest profit - this is an extension of the method we will discuss in a bit; called the simplex method) Any case, back to our two equations x + y = 3 x - 2 y = 4 What values of x and y satisfy these equations Two methods a) graphing y = 3 - x y = x/2-2 x around ; y around -.2 to -.5 good for fast approximations b) substitution

3 Phys_108B_Linear_Eqn. Page 3 x + y = 3 x - 2 y = 4 rearrange first y = 3 - x substitute in second x - 2 y = 4 x - 2 (3 - x) = 4 x x = 4 3x = 10 x = 10/3 = 3 1/3 back into rearranged first y = 3 - x = 3-3 1/3 = - 1/3 gives exact solution Problem is that as soon as one gets more than 2 variables (x, y and z) both these methods have real practical problems graphical - have to do multidimensional graphs substitution - gets real clumsy real fast and sometimes one can have independent variables, so a better method for solving was needed In the 1800's several methods of solving sets of linear equations were developed and we will look briefly, at one method and then look at how a program such as MathView can solve it A few definitions A matrix is a rectangular array of numbers and can be used to write simultaneous linear equations a 11 x 1 + a 12 x 2 + a 13 x 3 = b 1

4 Phys_108B_Linear_Eqn. Page 4 a 21 x 1 + a 22 x 2 + a 23 x 3 = b 2 a 31 x 1 + a 32 x 2 + a 33 x 3 = b 3 To get the equations from the matrix, you multiply the elements of first row of the matrix by the corresponding element in the vertical column The second equation is the second row of the matrix times the vertical column And so on. The array of constants is the matrix and a determinant is a particular calculated value of the matrix and the determinant is used in solving the linear equations. The determinant of a square matrix (that is, one that has as many rows as columns, which means that we, in the case of simultaneous linear equations, has as many independent

5 Phys_108B_Linear_Eqn. Page 5 equations as we have variables) is the sum of the product of the downward diagonals minus the sum of the product of the upward diagonals. Yeah, right. 2 x = 3*9-7*5 = = suppose had 3 5 = 3*10-6*5 = = nd row is twice the first (not independent) so the determinant of a matrix of non independent equations will equal 0. [Necessary but not sufficient; can equal zero for other cases] 3 x 3 - add copies of the first two columns and only evaluate the full diagonals

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7 Phys_108B_Linear_Eqn. Page 7 Homework = evaluate

8 Phys_108B_Linear_Eqn. Page 8 The determinants of larger square matrices are evaluated somewhat differently by a method called cofactoring, which is discussed in USED MATH, page 118 (in my office if interested). you want to look you want to check it oiu Now, on to our simultaneous equations. In the 1800's a mathematician named Cramer found that one could use the determinants of matrices to solve them. We are interested in the results, not the proof and even the results are cumbersome. A matrix has only one determinant, so what one does is for the variable, say, x 1, make a new matrix by replacing the a 1i coefficients by the b 1 's, evaluate the determinant for this and then divide the result by the determinant of the original matrix to get our value. Let's just do our 2 x 2 from above

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10 Phys_108B_Linear_Eqn. Page 10 solve x - 2 y = 3 2x - 5 y = 7 by this method Now, all of this is laborious, but it is a lot easier than substitution. And one would never do it by hand now adays. In fact, one of the first things computers were put to use for was solving sets of simultaneous linear equations during World War II. So we are going to look at how MathView does it, but other programs work as well (my calculator). Here is a simple way of substitution, works for 2 equations and is fairly fast

11 Phys_108B_Linear_Eqn. Page 11 For larger systems, you use MathView's Matrix Operators You can either enter the matrix by using the matrix button at the top or by hand. We will do the example from the book 5x + 6y - z + w = 3 2x + 2y + 4z - w = 7-4x + 3y + 9z + 12w = -4 y + 3z - w = 0 In MathView you open a new notebook and, as the first proposition you enter

12 Phys_108B_Linear_Eqn. Page 12 As you do this, the matrix appears. You then isolate the columns containing the variables (x, y,z,w) by highlighting, clicking the mouse while holding down the command (apple) key and dragging it onto the proposition marker (square); highlight the numerical side, expand (twice) and calculate - a quick as boiled asparagus, there are your answers

13 Phys_108B_Linear_Eqn. Page 13 Here is our little 2 x 2 from way back when

14 Phys_108B_Linear_Eqn. Page 14 Homework 1) Evaluate the determinants of the following matrices ) solve x - 2 y = 3 2x - 5 y = 7 by using determininants You may use Mathview or do by hand. 3) Solve using MathView the equations x + y + z = 10 2x + 3y + 5z = -20 x - 2y - 3z = 100

15 Phys_108B_Linear_Eqn. Page 15 4) Four gizmos and three whatits cost $42 while three gizmos and sven whatits cost $37. What is the price of a gizmo and of a whatsit? In the future I will give you a circuit diagram; you can write Kirchoff's rules for it and solve for the unknows.