Category Archive for: Linear Algebric Equations

Ethanol Production

Ethanol Production Engineers in The food chemical industries use fermentation in many processes. The following equation describes Baker’s yeast fermentation. a (C6 HI 206) + b (02) + c (NH3) – C6 HI ON 03 + d (H 20) + e (C 02) + f (C 2 H 60) The variables a, b, …• f represent…

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An Electrical-Resistance Network

An Electrical-Resistance Network The circuit shown in Figure 6.2-1 has five resistances and two applied voltages. Assuming that the positive directions of current flow are in the directions shown in the figure, Kirchhoff’s voltage law applied to each loop in the circuit gives -VI + Rii, + R4i4 = 0 -R4i4 + R2i2 + R5i5 = 0…

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Left-Division Method with Three Unknowns

Left-Division Method with Three Unknowns Use the left-division method to solve the following set: 3x + 2y – 9z = -65 -9x-5y+2z=16 6x +Ty +3z = 5 • Solution The matrix A is We can use MATLAB to check the determinant of A to see whether the problem is singular. The session looks like this:…

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Gauss Elimination

Gauss Elimination Solve the following set using Gauss elimination -x + y + 2z = 2 3x – y + z = 6 -x+3y+4z =4 • Solution The solution proceeds as follows: 1. Equation (6.1-1) is the pivot equation. Multiply it by -I and add the result to (6.1-3) to obtain 2y + 2z = 2,…

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Summary

Summary Once you have finished this chapter, you should be able to solve by hand systems of linear algebraic equations that have few variables and use MATLAB to solve1 systems that have many variables. If the number of equations in the set equalthe number of unknown variables, the matrix A is square and MATLAB provides two ways of…

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Overdetermined Systems

Overdetermined Systems An over-determined system is a set of equations that has more independent equations than unknowns. For such a system the matrix inverse method and Cramer’s method will not work because the A matrix is not square. However, some over-determined systems have exact solutions, and they can be obtained with the left-division method x = A/b.For other over-determined…

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Underdetermined Systems

Underdetermined Systems You have seen how to use the matrix inverse method x = A -1b to solve the equation set Ax = b. However, this method works only if the matrix A is square; that is, if the number of unknowns equals the number of equations. Even if A is square, the method will not work if…

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Cramer’s Method

Cramer’s Method Cramer’s method is a systematic method for solving equations, but it is not used as a basis for computer packages because it can be slow and very sensitive to numerical round-off error, especially for a large number of equations. We introduce, it here to gain insight into the requirements for a set of equations to have a…

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Matrix Methods for Linear Equations

Matrix Methods for Linear Equations Sets of linear algebraic equations can be expressed in matrix notation, a standard and compact method that is useful for expressing solutions and for developing software applications with an arbitrary number of variables. This section describes the use of matrix notation. As you saw in Chapter 2, a matrix is an ordered array…

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Elementary Solution Methods

Elementary Solution Methods You are sure to encounter situations in which MATLAB is not available (such as on a testl), and thus you should become familiar with the hand-solution methods. In addition, understanding these methods will help you understand the MATLAB responses and the pitfalls that can occur when obtaining a computer solution. Finally, hand…

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