thursday check out calculator if you don’t have one
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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!
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