Download Section 4-6:Matrices

Monday
• Pick up Note Sheet in
Front
• Check Out calculator if
you don’t have one.
Warm-up
• Solve this system of equations by the Elimination
method, then graphing
(7,  6)
3 x  y  15
 x  2 y  19
Section 3-4: Solving Systems of
Equations with Matrices
Pages 178- 185
Objectives
• I can solve systems of linear equations
containing 3 variables using matrices
• I can write matrix equations from a system
of linear equations
• I can write system of equations and solve
for word problems
Matrix Equations
• First the equations must have all variables
in the same order and all to the left of the
equation sign. The constant must be to the
right of the equation sign.
• ALWAYS double check this part of the
problem.
Matrix Vocabulary
• Every matrix is made up of N – rows and Mcolumns. So is called a N x M matrix
• Common matrices are 2x2, 3x3, 2x1, and 3x1 for
this section; however can be as large as you like.
• 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:
7x + 5y = 3
3x – 2y = 22
• We will make 3 matrices to make the Matrix
Equation:
7 5 
 3  2 


Matrix A
 x  3 
 y   22
   
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 
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.
Solving the Matrix Equation
• Follow along on the calculator
instructions. (Handout)
• We are going to enter the two
required matrices: A and B
2nd
x
1
This takes you to MATRIX MODE
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]
2nd
x
1
This takes you to back to MATRIX
MODE
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)
Now perform calculation 2nd x-1 then ENTER
Now x-1, 2nd x-1 then arrow down to [B], then ENTER
Now ENTER to find solution
Solution is: (-3, 7)
Your turn
 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)
Problem #3
• At a pizza shop, two small pizzas, a liter of
soda, and a salad cost $14; one small pizza,
a liter of soda, and three salads cost $15;
and three small pizzas and a liter of soda
cost $16. What is the cost of each item sold
separately?
Problem #3 Solution
• At a pizza shop, two small pizzas, a liter of soda, and a
salad cost $14; one small pizza, a liter of soda, and three
salads cost $15; and three small pizzas and a liter of soda
cost $16. What is the cost of each item sold separately?
•
•
•
•
•
•
Assign variables:
P – small pizza; L – liter of soda; S- salad
Now write equations:
2P + 1L + 1S = 14
1P + 1L + 3S = 15
3P + 1L = 16
• Solution: (5, 1, 3)
Homework
• Matrix Worksheet
• Don’t forget, ONE wrong keypunch and
you get the wrong answer!!
• Watch out for Negative Numbers!