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

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8.1 Matrix Solutions to Linear
SystemsVeronicaFangzhuXing3rd 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 systems solution.

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• Example 3
• Use matrices to solve the system
• 3xy2z31
• xy2z19
• x3y2z25
• 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 systems 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
• Get 1 in the upper left-hand corner
• Use the 1 in the first column to get 0s below it
• Get 1 in the second row, second column.
• Use the 1 in the second column to make the
  remaining entries in the second column 0
• Get 1 in the third row, third column.
• Use the 1 in the third column to make the
  remaining entries in the third column 0.
• Continue this procedure as far as possible.
• 3. Use the reduced row-echelon form of the matrix
  step 2 to write the systems solution set.(
  back-substitution is not necessary)

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• Example 4
• Use Gauss-Jordan elimination to solve the system
• 3xy2z31
• xy2z19
• x3y2z25