Download Section 4-6:Matrices

Thursday
• Make-up Quiz #1
before next class
• Submit an assignment
for a late grade to clear
a zero
Solving Systems of Equations
with Matrices
Objectives
• I can write matrix equations from a system
of linear equations
• I can solve systems of linear equations
containing 3 variables using matrices
• I can solve for word problems with
matrices
Matrix Equations
• Must have all variables in the same order
and all to the left of the equation sign. The
constant number must be to the right of the
equation sign.
• ALWAYS double check this part of the
problem.
Systems of Equations with 3
Variables
• The solution is always a triple ordered pair
(x, y, z).
• You may again have one solution, no
solutions, or infinite solutions.
Matrix Size
•
•
•
•
•
Every matrix is made up of:
M – rows
N- columns
So is called a M x N matrix
The following matrix is 3x3 because it has 3
rows and 3 columns
3 4
2
1

5
7


 5  7 5
What Size?
 2 3
 1 4


2X2
2
 4 
 
 5 
3X1
 3 4  What Size?
5 7


4
X
2
 2 5 


 8 12 
 4 
5
 
2X1
Converting Equations into Matrices
• Given the following linear equations:
2x + y – z = 5
3x – 2y + z = 16
4x + 3y – 5z = 3
• 3 matrices to make the Matrix Equation:
 2 1 1
 3 2 1  


 4 3 5
Matrix A
 x
 y 
 
 z 
5
16 
 
 3 
Matrix B
Example 2: Matrix Equations
• Given these equations, write a matrix equation:
3x + 4y + 2z = -9
3y – 5z = 12
2x – y = 5
3x + 4y + 2z = -9
0x + 3y – 5z = 12
2x –1y + 0z = 5
• Anytime a variable is missing, put a ZERO for its
place. It’s always best to rewrite the equations
with all terms before writing the matrix equation.
2   x    9
3 4
0 3  5   y    12 

    
2  1 0   z   5 
Calculator Steps
• 1. Enter Matrix A data
• 2. Enter Matrix B data
• 3. Perform this calculation :
1
[ A] [ B]
Example:
Solving the Matrix Equation
2 3 

1 2


Matrix A
 x   15 

 y   17 
  

Matrix B
MATRIX MODE
2nd
x
1
Arrow over to EDIT
With [A] selected hit ENTER
Type in Matrix [A] Dimensions 2 x 2,
then ENTER
Enter the data for Matrix [A]
GO back to MATRIX MODE
2nd
x
1
Arrow over to EDIT and DOWN to
Matrix [B]
Hit ENTER and then Matrix [B] size 2 x 1,
then ENTER
Type in data for Matrix [B], then 2nd MODE (quit)
Next get a blank screen
2nd
MODE
Calculate Solution
1
A B
Now ENTER to find solution
Solution is: (-3, 7)
Your turn
Solve by Matrices
x  2 y  4 z  19
2 x  y  3 z  14
3x  y  2 z  5
x  2 y  4 z  19
2 x  y  3 z  14
3x  y  2 z  5
 1 2 4 


2
1

3


3 1 2 



x
 y
 
 z 

 19 
 14 


 5 
(1, 6, -2)
Limitations
•
•
•
•
•
No Solution and Infinite Solutions
Matrices will NOT solve
You get an Error Message
“Singular Matrix”
You will have to look at slopes and yintercepts
Word Problems
• Highlight the key information
• Assign variables to represent the unknown
values
• Write equation to reflect the data.
Problem #1
• You have two jobs. One as a lifeguard and
one as a cashier. Your lifeguard job pays
$8 per hour and cashier pays $6 per hour.
Last week you worked a total of 14 hours
between the two jobs and earned $96. How
many hours did you work at each job?
Problem #1 Solution
• You have two jobs. One as a lifeguard and one as a
cashier. Your lifeguard job pays $8 per hour and cashier
pays $6 per hour. Last week you worked a total of 14
hours between the two jobs and earned $96. How many
hours did you work at each job?
•
•
•
•
•
Assign variables:
L – Hours at lifeguard; C – hours at cashier
Now write equations:
L + C = 14
8L + 6C = 96
• Solution: (6, 8)
Problem #2
• During a single calendar year, a state
trooper issued 375 citations for warnings
and speeding violations. There were 37
more warnings than speeding violations.
How many of each citation were issued?
Problem #2 Solution
• During a single calendar year, a state trooper issued 375
citations for warnings and speeding violations. There
were 37 more warnings than speeding violations. How
many of each citation were issued?
•
•
•
•
•
Assign variables:
W – # of warnings; S - # of speeding
Now write equations:
W + S = 375
W = S + 37
• Solution: (206, 169)
Homework
Worksheet 6-5