Section 4-6 Using Matrices to Solve Systems of Equations SOL AII.12

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Section 4-6Using Matrices to Solve Systems of
EquationsSOL AII.12
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Matrix Equations

• We have solved systems using graphing, but now we
  learn how to do it using matrices. This will be
  particularly useful when we have equations with
  three variables.

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Matrix Equation

• Before you start, make sure that both of your
  equations are in standard form and the variables
  are in the same order (alphabetical usually is
  best).

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Setting up the Matrix Equation

• Given a system of equations -2x - 6y
  0 3x 11y 4
• Since there are 2 equations, there will be 2
  rows.
• Since there are 2 variables, there will be 2
  columns.

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Setting up the Matrix Equation

• There are 3 parts to a matrix equation
• 1)The coefficient matrix,
• 2)the variable matrix, and
• 3)the constant matrix.

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-2x - 6y 03x 11y 4

• The coefficients are placed into the coefficient
  matrix.

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-2x - 6y 03x 11y 4

• Your variable matrix will consist of a column.

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-2x - 6y 03x 11y 4

• The matrices are multiplied and represent the
  left side of our matrix equation.

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-2x - 6y 03x 11y 4

• The right side consists of our constants. Two
  equations two rows.

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-2x - 6y 03x 11y 4

• Now put them together.

Well solve it later!
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Create a matrix equation

• 3x - 2y 7 y 4x 8
• Put them in Standard Form.
• Write your equation.

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Create a matrix equation

• 3a - 5b 2c 9
• 4a 7b c 3
• 2a - c 12

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Solving a matrix equation

• To solve matrix equations, get the variable
  matrix alone on one side.
• Get rid of the coefficient matrix by multiplying
  by its inverse

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• When solving matrix equations we will always
  multiply by the inverse matrix on the left of the
  coefficient and constant matrix. (remember
  commutative property does not hold!!)

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• The left side of the equation simplifies to the
  identity times the variable matrix. Giving us
  just the variable matrix.

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• Using the calculator we can simplify the left
  side. The coefficient matrix will be A and the
  constant matrix will be B. We then find A-1B.

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• The right side simplifies to give us our answer.
• x -6
• y 2
• You can check the systems by graphing,
  substitution or elimination.

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Advantages

• Basically, all you have to do is put in the
  coefficient matrix as A and the constant matrix
  as B. Then find A-1B. This will always work!!!
• NO SOLVING FOR Y!!!!! )

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Solve

• Plug in the coeff. matrix as A
• Put in the const. matrix as B
• Calculate A-1B.

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Solve

• r - s 3t -8
• 2s - t 15
• 3r 2t -7

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Assignment Solving Matrix Equations

• Wkst 4-6 1 - 8
• pg. 224
• 21 - 26 all

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Word Problem Systems

• pg. 170 25
• The sum of three numbers is 12. The 1st is 5
  times the 2nd. The sum of the 1st and 3rd is 9.
  Find the numbers.

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Word Problem Systems

• The sum of three numbers is 12.
• x y z 12
• The 1st is 5 times the 2nd.
• x 5y
• The sum of the 1st and 3rd is 9.
• x z 9

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Word Problem Systems

• x y z 12 x 5y gt x - 5y 0 x z
  9

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Word Problem Systems

• x y z 12 x - 5y 0 x z 9