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Chapter 2 Simultaneous Linear Equations

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A system of m linear equations in n variables is a set of m equations, each of which is linear in the same n variables: A solution is a set of scalars x 1, x 2, …, x n that when substituted in the system satisfies the given equations. A linear system can possess exactly one solution, an infinite number of solutions, or no solution. A linear system is called consistent if it has at least one solution and inconsistent if it has no solution. A linear system can be written in matrix form: Ax = b (see details on the board) A linear system is called homogenous if b=0 2.1 Linear systems

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A manufacturer produces desks and bookcases. Desks d require 5 hours of cutting time and 10 hours of assembling time. Bookcases b require 15 minutes of cutting time and one hour of assembling time. Each day, the manufacturer has available 200 hours for cutting and 500 hours for assembling. The manufacturer wants to know how many desks and bookcases should be scheduled for completion each day to utilize all available workpower. Show that this problem is equivalent to solving two equations in the two unknowns d and b. 2.1 Linear systems (HW example) Modeling a real-life situation as a linear model

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Take the first equation and solve for x 1 in terms of x 2, …, x n and then substitute the value of x 1 into all the other equations, thus eliminating it from those equations. This new form is the first derived set. Working with the first derived set, solve the second equation for x 2 in terms of x 3, …, x n and then substitute this value of x 2 into the third, fourth, etc. equations, thus eliminating it. Do this process recursively with other variables. The resulting system can be solved by back substitution. An example on the board. We will consider in more details more advanced methods which use matrices. 2.2 Solutions by Substitution

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The matrix derived from the coefficients and constant terms of a system of linear equations is called the augmented matrix of the system. The matrix containing only the coefficients of the system is called the coefficient matrix of the system. System Augmented Matrix Coefficient Matrix x y z const. Augmented matrix of a linear system

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(E1) Interchange any two rows. (E2) Multiply any row by a nonzero scalar. (E3) Add a multiple of a row to another row. Two matrices are said to be row-equivalent if one can be obtained from the other by a finite sequence of elementary row operations. Row-equivalent systems have the same set of solutions. Elementary row operations

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Gaussian Elimination Write the augmented matrix of the system. Use elementary row operations to transform it to an equivalent row-reduced form. (this is most often accomplished by using (E3) with each diagonal element to create zeros in all columns directly below it, beginning with the first column) The system associated with row-reduced matrix can be solved easily by back- substitution.

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Linear SystemAssociated Augmented matrix R2+R1R2R2+R1R2 (2)(2) R3+R2R3R3+R2R3 R 3 R 3 Gaussian Elimination: Example 1

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A system with no solution Solve the system (1)(1) (2)(2) (3)(3) 0 = 2 … ??? The original system of linear equations is inconsistent. Gaussian Elimination: Example 2

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