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

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

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


1
Section 4-6 Using Matrices to Solve Systems of
Equations SOL AII.12
2
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.

3
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).

4
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.

5
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.

6
-2x - 6y 0 3x 11y 4
  • The coefficients are placed into the coefficient
    matrix.

7
-2x - 6y 0 3x 11y 4
  • Your variable matrix will consist of a column.

8
-2x - 6y 0 3x 11y 4
  • The matrices are multiplied and represent the
    left side of our matrix equation.

9
-2x - 6y 0 3x 11y 4
  • The right side consists of our constants. Two
    equations two rows.

10
-2x - 6y 0 3x 11y 4
  • Now put them together.

Well solve it later!
11
Create a matrix equation
  • 3x - 2y 7 y 4x 8
  • Put them in Standard Form.
  • Write your equation.

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

13
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

14
  • 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!!)

15
  • The left side of the equation simplifies to the
    identity times the variable matrix. Giving us
    just the variable matrix.

16
  • 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.

17
  • The right side simplifies to give us our answer.
  • x -6
  • y 2
  • You can check the systems by graphing,
    substitution or elimination.

18
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!!!!! )

19
Solve
  • Plug in the coeff. matrix as A
  • Put in the const. matrix as B
  • Calculate A-1B.

20
Solve
  • r - s 3t -8
  • 2s - t 15
  • 3r 2t -7

21
Assignment Solving Matrix Equations
  • Wkst 4-6 1 - 8
  • pg. 224
  • 21 - 26 all

22
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.

23
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

24
Word Problem Systems
  • x y z 12 x 5y gt x - 5y 0 x z
    9

25
Word Problem Systems
  • x y z 12 x - 5y 0 x z 9
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