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8.1 Matrix Solutions to Linear Systems Veronica Fangzhu Xing 3 rd period

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Solving Linear System Using Matrices An augmented matrix has a vertical bar separating the columns of the matrix into two groups The coefficients of each variable -------- the left of the vertical line The constants---------right ( if any variable is missing, its coefficient is 0) x +2y -5z =-19 y +3z =9 z =4

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Matrix Row Operations

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Solving linear System Using Gaussian Elimination Write the augmented matrix for the system. Write the system of linear equations corresponding to the matrix in step 2 and use back-substitution to find the system’s solution.

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Example 3 : Use matrices to solve the system: 3x+y+2z=31 x+y+2z=19 x+3y+2z=25 Step 1 : Write the augmented matrix for the system.

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Step 2 : Use matrix row operations to simplify the matrix to row-echelon form, with 1s down the diagonal from upper left to lower right, and 0s below the 1s.

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Step 3 : Write the system of linear equation corresponding to the matrix in step 2 and use back-substitution to find the system’s solution.

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Solving linear system Using Gauss Jordan Elimination 1. Write the augmented matrix for the system. 2. Use matrix row operations to simplify the matrix to a row-equivalent matrix in reduced row-echelon form, with1sdown the main diagonal from upper left to lower right, and 0s above and below the 1 s a)Get 1 in the upper left-hand corner b)Use the 1 in the first column to get 0s below it c)Get 1 in the second row, second column. d)Use the 1 in the second column to make the remaining entries in the second column 0 e)Get 1 in the third row, third column. f)Use the 1 in the third column to make the remaining entries in the third column 0. g)Continue this procedure as far as possible. 3. Use the reduced row-echelon form of the matrix step 2 to write the system’s solution set.( back-substitution is not necessary)

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Example 4 : Use Gauss-Jordan elimination to solve the system 3x+y+2z=31 x+y+2z=19 x+3y+2z=25

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