United Arab Emirates University
College of Engineering &
College of Science
Math 2220: Linear Algebra and
Prepared by: Dr. Mousa Hussein
ENGINEERING APPLICATIONS OF SYSTEMS OF LINEAR
Many “real life” situations are governed by a system of differential equations. Physical problems usually have more than one dependent variable to be considered. Models of such problems can give rise to system of differential equations in which there are two or more dependent variable and one independent variable. Because we are going to be working almost exclusively with systems of equations in which the number of unknowns equals the number of equations we will restrict our discussion to these kinds of systems.
Let‟s start with the following system of n equations with the n unknowns, x1, x2,…, xn
Note that in the subscripts on the coefficients in this system, aij, the i corresponds to the equation that the coefficient is in and the j corresponds to the unknown that is multiplied by the coefficient. This can be put in a simple linear equation form
ax b ,
Where a and b are given real or complex numbers and x is an unknown. To use linear algebra to solve this system we will first write down the augmented matrix for this system. An augmented matrix is really just the all the coefficients of the system and the numbers for the right side of the system written in matrix form. Here is the augmented matrix for this system. However in our analysis we will use Matlab to help us solve such system.
We will start with the simplest possible linear equation ax b , such equation has three possible solutions.
a) If a 0 and b 0 , then 0 x 0 b which is false for any value of x, and so there are no solutions
b) If a 0 , then the equation has a unique solution x b for any value of b. a
Prepared by: Dr. Mousa Hussein
c) If a b 0 , then 0 x 0 for any value of x and so there are infinitely many solutions.
General Strategy for Solving Systems of Linear Equations:
Using Mathematic to solve an applied problem involves translation of the features of the problem into mathematical language (terminology, symbols, equations and so on). We refer to this translation process as building a mathematical model of the problem. In all applications of linear equations, we will follow the same general strategy First: Identify and label the unknowns.
In other words, what are we asked to find? In answering this question, you should note down something like the following:
Let x be the number of video games.
Let y be the number of applications.
Let z be the number of documents.
Note that all the unknowns should be numbers, so we should not say something like "Let x = video games."
Second: Use the information given to set up equations in the unknowns. How to do this depends on the way the problem is worded. We will look at a few examples below to develop some strategies.
Third: Solve the system to obtain the values for the unknowns. We will use the help of Matlab to obtain solve for the unknowns. There are several kinds of applications generally found in textbooks. Applications in which the given information can be tabulated Applications in which some of the given information must be translated from words into equations.
Applications of specialized types, such as "transportation problems" and "traffic flow problems." These require special techniques for setting up the system of equations.
We will start with simple applications and then we will move on into more specialized Engineering applications.
A Yogurt Company makes three yogurt blends: LimeOrange, using 2 quarts of lime yogurt and 2 quarts of orange yogurt per gallon; LimeLemon, using 3 quarts of lime yogurt and 1 quart of lemon yogurt per gallon; and OrangeLemon, using 3 quarts of orange yogurt and 1 quart of lemon yogurt per gallon. Each day the company has 800 quarts of...
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