Section 3.6

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Section 3.6 Solving Systems Using Matrices

• Students will be able to
• Represent a system of linear equations with a
  matrix
• To solve a system of linear equations using
  matrices
• NEW VOCABULARY
• Matrix
• Matrix Element
• Row Operations

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Section 3.6 Solving Systems Using Matrices
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Section 3.6 Solving Systems Using Matrices

• An array of numbers, such as each of those
  suggested by the tile arrangements in the
• Solve It, is a matrix.
• Essential Understanding
• You can use a matrix to represent and solve a
  system of equations without writing ANY
  variables.
• REAL LIFE

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Section 3.6 Solving Systems Using Matrices

• A matrix is a rectangular array of numbers. You
  usually display the array with brackets. The
  dimensions of a matrix are the number of rows and
  columns in the array.
• Each number is a matrix is a matrix element. You
  can identify a matrix element by its row and
  column of numbers. In Matrix A, a12 is the
  element in row 1 and column 2. a12 is the element
  4.

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Section 3.6 Solving Systems Using Matrices

• You can represent a system of equations
  efficiently with a matrix.
• Each matrix row represents an equation.
• The last matrix column shows the constants to the
  right of the equal signs.
• Each of the other columns shows the coefficients
  of one of the variables.

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Section 3.6 Solving Systems Using Matrices
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Section 3.6 Solving Systems Using Matrices

• Problem 1
• How can you represent the system of equations
  with a matrix?
• 2x y 9
• x 6y -1

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Section 3.6 Solving Systems Using Matrices

• Problem 1
• How can you represent the system of equations
  with a matrix?
• x 3y z 6
• x 3z 12
• y -5x 1

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Section 3.6 Solving Systems Using Matrices

• Problem 1
• How can you represent the system of equations
  with a matrix?
• -4x 2y 7
• 3x y -5

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Section 3.6 Solving Systems Using Matrices

• Problem 1
• How can you represent the system of equations
  with a matrix?
• 4x y 2z 1
• y 5z 20
• 2x -y 7

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Section 3.6 Solving Systems Using Matrices

• Problem 2
• What linear system of equations does this matrix
  represent?

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Section 3.6 Solving Systems Using Matrices

• Problem 2
• What linear system of equations does this matrix
  represent?

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Section 3.6 Solving Systems Using Matrices

• You can use a matrix that represents a system of
  equations to solve the system. In this way, you
  do not have to write the variables.
• To solve the system using a matrix, use the steps
  for solving by elimination.
• Each step is a row operation.

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Section 3.6 Solving Systems Using Matrices

• Your goal is to use row operations to get a
  matrix in the form
• Notice that the first matrix represents the
  system x a, y b, which then will be the
  solution of a system of two equations in two
  unknowns. The second matrix represents the
  system x a, y b, and z c.

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Section 3.6 Solving Systems Using Matrices
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Section 3.6 Solving Systems Using Matrices

• Problem 3
• What is the solution of the system?
• x 4y -1
• 2x 5y 4

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Section 3.6 Solving Systems Using Matrices

• Problem 3
• What is the solution of the system?
• x y 5
• -2x 4y 8

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Section 3.6 Solving Systems Using Matrices

• Problem 3
• What is the solution of the system?
• x 3y 22
• 2x - y 2

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Section 3.6 Solving Systems Using Matrices

• Matrices that represent the solution of a system
  are in reduced row echelon form. Many
  calculators have a rref(reduced row echelon form)
  function for working with matrices.
• This function will do all the row operations for
  you. You can use the rref to solve a system of
  equations.

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Section 3.6 Solving Systems Using Matrices

• What is the solution of the system of equations?
• 2a 3b c 1
• -4a 9b 2c 8
• -2a 2c 3

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Section 3.6 Solving Systems Using Matrices

• What is the solution of the system of equations?
• a 4b 6c 21
• 2a 2b c 4
• -8b c -1

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Section 3.6 Solving Systems Using Matrices