Warm-Up Find the x, y and z intercepts:a)

Solve this 2-D system by Graphing on your calculator

Solving Systems of Equations

Learning Targets

Refresher on solving systems of equations Matrices

– Operations– Uses– Reduced Row Echelon Form

Solving Systems of Equations

There are multiple ways to solve systems of equations:– Graphing– Substitution (Equal Values Method)– Elimination

Solve the System by Graphing

f(x)=(2/3)x+15

f(x)=-(4/5)x+2

Series 1

-19-18-17-16-15-14-13-12-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

-19-18-17-16-15-14-13-12-11-10-9-8-7-6-5-4-3-2-1

123456789

10111213141516171819

x

y

(-8.86,9.09)

Solve the System using Algebra

Algebra Method cont.

Elimination Method:

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.

Matrix EquationBefore you start, make sure:

1. That all of your equations are in standard form.

2. The variables are in the same order (alphabetical usually is best).

3. If a variable is missing use zero for its coefficient.

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.

There are 3 parts to a matrix equation

1)The coefficient matrix,

2)the variable matrix, and

3)the constant matrix.

Setting up the Matrix Equation

-2x - 6y = 03x + 11y = 4

The coefficients are placed into the coefficient matrix.

2 6

3 11

-2x - 6y = 03x + 11y = 4

Your variable matrix will consist of a column.

x

y

-2x - 6y = 03x + 11y = 4

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

x

y

2 6

3 11

-2x - 6y = 03x + 11y = 4

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

0

4

-2x - 6y = 03x + 11y = 4

Now put them together.

2 6

3 11

x

y

0

4

We’ll solve it later!

Create a matrix equation 3x - 2y = 7

y + 4x = 8 Put them in Standard Form. Write your equation.

3 2

4 1

x

y

7

8

3a - 5b + 2c = 9

4a + 7b + c = 3

2a - c = 123 5 2

4 7 1

2 0 1

a

b

c

9

3

12

Create 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

Solving a matrix equation

2 6

3 11

x

y

0

4

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!!)

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

x

y

2 6

3 11

1 0

4

2 6

3 11

1 2 6

3 11

x

y

2 6

3 11

1 0

4

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.

x

y

2 6

3 11

1 0

4

The right side simplifies to give us our answer.

x = -6 y = 2 You can check the systems by

graphing, substitution or elimination.

x

y

6

2

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!!!

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

3 2

4 1

x

y

7

8

x

y

2111 411

Solve: r - s + 3t = -8 2s - t = 15 3r + 2t = -7

1 1 3

0 2 1

3 0 2

r

s

t

8

15

7

r

s

t

3

8

1

Working with

Matrices on TI-83, TI-84

Source: Mathbits

Explore:• How many matrices does your calculator have?• Use the right arrow key to move to MATH. Scroll

down and find rref. We will use this key later.• Use the right arrow key once more to highlight

EDIT.

Step 1:  Go to Matrix                  (above the  x-1

key) 

Step 2:  Arrow to the right to             EDIT to allow for              entering the matrix. 

Press ENTER

Step 3:  Type in the dimensions (size) of your matrix and enter the elements (press ENTER).    

Step 4:  Repeat this process for              a different matrix..

Step 5: Arrow to the right to             EDIT and choose a              new name.

Step 6:  Type in the dimensions (size) of your matrix and enter the elements (press ENTER).    

Using Matrices to Solve Systems of

Equations: • 1.  (using the inverse coefficient matrix)

Write this system as a matrix equation and solve:  3x  + 5y = 7 and 6x - y = -8

• Step 1:  Line up the x, y and              constant values.

• 3x  + 5y =  76x   -   y = -8

• Step 2:  Write as equivalent               matrices.

• Step 3:  Rewrite to separate out              the variables. 

Step 4:  Enter the two numerical matrices in the calculator.

Step 5:  The solution is obtained by multiplying both sides of the equation by the inverse of the matrix which is multiplied timesthe variables.

• Step 6:  Go to the home screen and enter the right side of the previous equation.

• The answer to the system, as seen on the calculator screen, is  x = -1 and y = 2.

Method 2• 2.  (using Gauss-Jordan elimination method with

reduced row echelon form )     Solve this system of equations:

• 2x - 3y + z = -54x -  y - 2z = -7-x + 2z = -1

• Step 1:  Line up the variables and              constants

• 2x - 3y  + z = -54x -  y  - 2z = -7

-x +0y + 2z = -1

• Step 2:  Write as an augmented              matrix and enter into              calculator.

• Step 3:  From the home screen, choose the rref function.  [Go to Matrix  (above the  x-1 key), move right→MATH, choose B: rref]

• Step 4:  Choose name of matrix             and hit ENTER

• Step 5:  The answer to the system, will be the last column on the calculator screen:          x = -3          y = -1          z = -2.

Method 2:• Case 1: Unique

solution Enter as a 3X4 matrix

Diagonal is all ones so there is a solution:

Method 2:• Case 2: No solution

Enter as a 3X4 matrix

Last row:

0 0 0 1No

solution.

Method 2:• Case 3: Infinitely Many Solutions

Enter as a 3X4 matrix

Last row:

0 0 0 0Infinitely Many Solutions

Website to Visualize the Solutions

• http://www.cpm.org/flash/technology/3dsystems.swf

For Tonight

• Intro to Matrices Worksheet