Systems of Linear Equations

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Systems of Linear Equations

• Lets say you need to solve the following for x,
  y, z
• 2x y 2z 10
• 3x 2y 2z 1
• 5x 4y 3z 4
• Two methods
• Gaussian elimination
• Cramers rule

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Gaussian Elimination

• For any system of independent linear equations,
• we can set up the following augmented matrix
• and perform elementary row operations to reduce
  it to row-echelon form

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Our Example

1. Multiply row 1 by 0.5
2. Multiply row 1 by -3 and add to row 2

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Our example (cont.)

1. Continue until we have row-echelon form

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Our example (concl.)

• This corresponds to
• ____________________ ____
• _____________ ____
• ______ ____

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Alternative Method Cramers Rule

1. Convert A to Ai where Ai are the matrices
   obtained by replacing the ith column with B

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Cramers Rule (cont.)

• Find the determinants of each of these matrices
• D det(A) _______
• N1 det(A1) _______
• N2 det(A2) _______
• N3 det(A3) _______

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Cramers Rule (concl.)

• The unique solution is now found by
• x N1/D _______
• y N2/D _______
• z N3/D _______

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Cramers Rule Works If and Only If

• Number of equations number of unknowns
• D ? 0

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Homework

• Solve each of the following systems of linear
  equations,
• a) using Gaussian Elimination
• b) using Cramers Rule
• 2x y z 3
• x y z 1
• x 2y 3z 4
• x 3y 2z 6
• 2x 4y 3z 8
• -3x 6y 8z -5