Ch2ppt - Systems of Linear Equations and Matrices Chapter 2 Systems of Linear Equations and Matrices n 2.2 Solution of Linear Systems by the

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Unformatted text preview: Systems of Linear Equations and Matrices Chapter 2 Systems of Linear Equations and Matrices n 2.2 Solution of Linear Systems by the Gauss-Jordan Method n 2.3 Addition and Subtraction of Matrices n 2.4 Multiplication of Matrices n 2.5 Matrix Inverses Systems of Linear Equations n A system of linear equations is a set of n linear equations in k variables (or unknowns) that are solved together. n The simplest linear system is one with 2 equations in 2 variables. n A solution of a system is a solution that satisfies all the equations in the system. Solving Systems of Linear Equations n Three methods n Graph the lines and identify the intersection (if any) n Substitution n Elimination n Example: 2 3 12 3 4 1 x y x y + =- = Graphing Method-8-6-4-2 2 4 6 8 10 12-9-6-3 3 6 9 12 15 18 2x+3y=12 3x-4y=1 ( x , y ) Substitution n Solve the first equation for y n Substitute this expression for y in the second equation. Solve for x 2 3 12 3 4 1 x y x y + =- = 4 3 3 12 2 2 y x x y =- =- 3 4 1 8 3 16 1 3 17 17 3 4 3 2 3 x x x x x x - = ÷ - + = = =- Substitution n Substitute x = 3 in either equation to solve for y . 2 3 12 3 4 1 x y x y + =- = ( 29 2 3 12 3 2 6 3 y y y + = = = Solution: ( 3 , 2 ) ( 29 ( 29 ( 29 ( 29 2 3 12 3 3 1 2 3 2 4 + =- = Elimination n In systems of equations where the coefficients of terms containing the same variable are opposites, the elimination method can be applied by adding the equations. If the coefficients of those terms are the same, the elimination method can be applied by subtracting the equations. 2 3 12 3 4 1 x y x y + =- = Elimination n Multiply the first equation by 4 and the second equation by 3, so the coefficients of y are negatives of each other. 2 3 12 3 4 1 x y x y + =- = 8 48 12 3 12 9 x x y y +- = = Elimination n Any solution of this system must also be the solution of the sum of the two equations n Substitute x = 3 in either equation to solve for y . 2 3 12 3 4 1 x y x y + =- = 8 48 9 3 1 1 1 2 7 5 2 1 x y x x y +- = = = 3 x = ( 29 2 3 12 3 2 6 3 y y y + = = = Solution: ( 3 , 2 ) 2.2 Gauss-Jordan Method n Useful when solving systems of equations with more than 2 equations and 2 variables. n Uses matrix representation to determine the solutions to a system of linear equations....
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This note was uploaded on 01/12/2012 for the course ECO 3401 taught by Professor Staff during the Fall '08 term at University of Central Florida.

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Ch2ppt - Systems of Linear Equations and Matrices Chapter 2 Systems of Linear Equations and Matrices n 2.2 Solution of Linear Systems by the

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