Find the quadratic function with y intercept (0,4) and passes through the points (2, 10) and ( 1, 7).

Transcription

1 Our objective is to solve problems such as... 1) My son, Jimmy, takes his coin jar into the bank. He pours all the coins into the "coin counting" machine. It reports that he had 273 coins (nickels and pennies) for a value of $8.13. How many of each coin did he have in his jar? 2) Find the quadratic function with y intercept (0,4) and passes through the points (2, 10) and ( 1, 7).

2 This is a system of equations. Our goal is to find values of x and y that make both equations true! Solving by graphing. Solution?

3 Recall the "elimination method" Solution?

4 Possible situations that come up when solving systems of equations. Independent exactly one solution Consistent 2x + y = 4 4x y = 8 Dependent infinitely many solutions 2x + y = 4 4x 2y = 8 Inconsistent no solutions 2x + y = 4 4x 2y = 6

5 Let's look at a very simple system... 2x + y = 6 y = 4 This process is called " substitution."

6 What we plan to do is solve systems of equations using a matrix. Why use matrices? Parts of a Matrix and the Order of Matrix A B C D What does a 3x2 matrix look like?

7 Augmented Matrix Starting System of Equations 2x + 4y = 8 Rewrite as Augmented matrix x 5y = 2 Starting System of Equations 3x + y = 0 Rewrite as Augmented matrix 2x + 6y = 3

8 Orginal system Augmented matrix Augmented matrix

9 Augmented matrix Row Echelon Form goal Ones on the diagonal! Zeros under the ones! Examples:

10 Ex 1 Row Echelon Form Find solution using " substitution" Ex 2 Row Echelon Form

11 Ex 1 Row Echelon Form Equations

12 Practice... Write each system as a matrix.

13 Practice... Back Substitution Augmented matrix in Row Echelon Form System of Equations Solve for x and y!

14 Augmented Matrix Gaussian Elimination (in your calculator) Starting System of Equations 2x + 4y = 8 2x + 7y = Rewrite as Augmented matrix Goal! Reduced to Row Echelon Form 3 Back Substitution (Solve) Calculator time!

15 Solve Using Gaussian Elimination Starting System of Equations 3x + 9y = 21 Rewrite as Augmented matrix Reduced to Row Echelon Form 2x 5y = 11 Back Substitution

16 Solve Using Gaussian Elimination Starting System of Equations Rewrite as Augmented matrix Reduced to Row Echelon Form 2x + 4y 6z = 6 x + y = 0 x + z = 3 Back Substitution

17 Extra Practice 2x + 4y 6z = 6 x + y = 0 x + z = 3

18 Our objective is to solve problems such as... 1) My son, Jimmy, takes his coin jar into the bank. He pours all the coins into the "coin counting" machine. It reports that he had 273 coins (nickels and pennies) for a value of $8.13. How many of each coin did he have in his jar?

19 2) Find the quadratic function with y intercept (0,4) and passes through the points (2, 10) and ( 1, 7).

20

21 Translate a word problem into a system of equations and solve. An investment of $3000 is placed into a combination of stocks and bonds. The annual rate of return is 4.5% for stocks and 8% for bonds. If the total annual return is $177, how much was invested in both?

22 A pharmacist has two vitamin supplement powders. The first powder is 25% vitamin B1 and 15% vitamin B2. The second is 15% B1 and 20% B2. How much of each powder should be used to produce a supplement that has grams of B1 and 120 grams of B2?

23 Find a quadratic equation that passes through the points: (2, 3), ( 2, 7), and (1, 2)

24 The following table shows the active chemical content of three different soil additives. Additive Ammonium Nitrate Phosphorus Iron 1 30% 10% 10% 2 40% 15% 10% 3 50% 5% 5% A soil chemist wants to prepare a soil product that combines some of each additive above in such a way that the produce has a total of 380 grams of ammonium nitrate, 95 grams of phosphorus, and 85 grams of iron. How much of each additive is required?

25 A goldsmith has two gold alloys. One alloy is 40% gold and the second allowy is 60% gold. How much of each should be used to produce 20 grams of an alloy that is 52% gold?

26 Adding Matrices Subtracting Matrices Scalar Multiplication Adding Matrices A B C + = Subtracting Matrices A B = C

27 Scalar Multiplication A A 3A = 3 =

28 Scalar Multiplication A = B= 2A B = 2A B =

29 Matrix Multiplication = Order

30

31

32

33 Matrix Multiplication A B C AB = Order AB =

34 More practice...

35 More practice...

36 Using Matrix Multiplication Rewrite a system of equations in matrix form. Coefficient matrix Variable matrix Constant matrix

37 Rewrite a system of equations in matrix form. Coefficient matrix Variable matrix Constant matrix

38 Translate a word problem into a system of equations and solve. An investment of $3000 is placed into a combination of stocks and bonds. The annual rate of return is 4.5% for stocks and 8% for bonds. If the total annual return is $177, how much was invested in both?

39 A pharmacist has two vitamin supplement powders. The first powder is 25% vitamin B1 and 15% vitamin B2. The second is 15% B1 and 20% B2. How much of each powder should be used to produce a supplement that has grams of B1 and 120 grams of B2?

40 A goldsmith has two gold alloys. One alloy is 40% gold and the second allowy is 60% gold. How much of each should be used to produce 20 grams of an alloy that is 52% gold?

41 Find a quadratic equation that passes through the points: ( 3, 0), ( 1, 6), and (1, 4)

42 Translate a word problem into a system of equations and solve. The following table shows the active chemical content of three different soil additives. Additive Ammonium Nitrate Phosphorus Iron 1 30% 10% 10% 2 40% 15% 10% 3 50% 5% 5% A soil chemist wants to prepare a soil product that combines some of each additive above in such a way that the produce has a total of 380 grams of ammonium nitrate, 95 grams of phosphorus, and 85 grams of iron. How much of each additive is required?

43 Translate a word problem into a system of equations and solve. The following table shows the active chemical content of three different soil additives. Additive Ammonium Nitrate Phosphorus Iron 1 30% 10% 10% 2 40% 15% 10% 3 50% 5% 5% A soil chemist wants to prepare a soil product that combines some of each additive above in such a way that the produce has a total of 380 grams of ammonium nitrate, 110 grams of phosphorus, and 90 grams of iron. How much of each additive is required?

44

45 Grams per 100 grams Food Type Carbs Fat Protein Prepare a diet that contains 23 grams of carbs, 18 grams of fat, and 39 grams of protein. Prepare a second diet that contains 35 grams of carbs, 28 grams of fat, and 42 grams of protein.

46