# Using%20Matrices%20to%20Solve%20Systems%20of%20Equations - PowerPoint PPT Presentation

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## Using%20Matrices%20to%20Solve%20Systems%20of%20Equations

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### Using Matrices to Solve Systems of Equations Matrix Equations We have solved systems using graphing, but now we learn how to do it using matrices. – PowerPoint PPT presentation

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Title: Using%20Matrices%20to%20Solve%20Systems%20of%20Equations

1
Using Matrices to Solve Systems of Equations
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.

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
• 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
Word Problem Systems
• 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.

22
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

23
Word Problem Systems
• x y z 12 x 5y gt x - 5y 0 x z
9

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