Thursday

• 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 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.

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.

Example: Solving the Matrix Equation

2 3

1 2

15

17

x

y

Matrix A Matrix B

12nd x

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

GO 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)

Next get a blank screen

2nd MODE

Calculate Solution

1A B

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!