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

3 15

2 19

x y

x y

(7, 6)

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 M- columns. 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

575

751

432

What Size?

2 3

1 4

2

4

5

2 X 2

3 X 1

What Size?3 4

5 7

2 5

8 12

4

5

4 X 2

2 X 1

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:

23

57

22

3

y

x

Matrix A Matrix B

Example 2: Matrix Equations

• Given these equations, write a matrix equation:3x + 4y + 2z = -9

3y – 5z = 12

2x – y = 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.

3x + 4y + 2z = -9

0x + 3y – 5z = 12

2x –1y + 0z = 5

5

12

9

012

530

243

z

y

x

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

12nd x 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]

12nd x 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

19

14

5

x

y

z

(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 y-intercepts

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!