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Solving System of Linear Equations

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1. Diagonal Form of a System of Equations 2. Elementary Row Operations 3. Elementary Row Operation 1 4. Elementary Row Operation 2 5. Elementary Row Operation 3 6. Gaussian Elimination Method 7. Matrix Form of an Equation 8. Using Spreadsheet to Solve System 2

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A system of equations is in diagonal form if each variable only appears in one equation and only one variable appears in an equation. For example: 3

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Elementary row operations are operations on the equations (rows) of a system that alters the system but does not change the solutions. Elementary row operations are often used to transform a system of equations into a diagonal system whose solution is simple to determine. 4

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Elementary Row Operation 1 Rearrange the equations in any order. 5

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Rearrange the equations of the system so that all the equations containing x are on top. 6

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Elementary Row Operation 2 Multiply an equation by a nonzero number. 7

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Multiply the first row of the system so that the coefficient of x is 1. 8

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Elementary Row Operation 3 Change an equation by adding to it a multiple of another equation. 9

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Add a multiple of one row to another to change so that only the first equation has an x term. 10

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Gaussian Elimination Method transforms a system of linear equations into diagonal form by repeated applications of the three elementary row operations. 1. Rearrange the equations in any order. 2. Multiply an equation by a nonzero number. 3. Change an equation by adding to it a multiple of another equation. 11

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Continue Gaussian Elimination to transform into diagonal form 12

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14 The solution is ( x,y,z ) = (4/5,-9/5,9/5).

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It is often easier to do row operations if the coefficients and constants are set up in a table (matrix). Each row represents an equation. Each column represents a variables coefficients except the last which represents the constants. Such a table is called the augmented matrix of the system of equations. 15

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Write the augmented matrix for the system 16 Note: The vertical line separates numbers that are on opposite sides of the equal sign.

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The three elementary row operations for a system of linear equations (or a matrix) are as follows: Rearrange the equations (rows) in any order; Multiply an equation (row) by a nonzero number; Change an equation (row) by adding to it a multiple of another equation (row). 17