MAT141, Chapter 2 Finite Mathematics MAT 141: Chapter 2 Notes Solutions to Linear Systems by the Echelon Method (i.e. elimination) David J. Gisch S ystems of Linear Equations One S olution 1
MAT141, Chapter 2 No S olution Infinite S olutions S olving a S ystem Using Elimination Investigating Rule 3 • Lets look at the following system. � � 4 �1� ▫ � � � � � 2 �2� • Below is the graph of these equations. They do intersect and have a solution of 2, 4 . • Try some different combinations using rule 3 and graph your results. For Example, I will take 2(1)+(2) which results in • Rule 1 seems clear. 2� � 8 • Rule 2 seems clear. � � � � � 2 3� � � � 10 • Rule 3 is ok? � � 1 3 � � 10 3 2
MAT141, Chapter 2 Using Transformations to S olve Using Transformations to S olve • Using these transformation rules we solve using a Example: Solve the system using elimination. process called elimination. �2� � 3� � 12 �1� ▫ You try to eliminate a variable to solve for another. �2� ▫ Then you substitute that answer back into an equation to find the 3� � 4� � 1 other. Example: Solve the system using elimination. ��4� � � � �7 �1� �2� � � � � �2 Using Transformations to S olve Using Transformations to S olve Example: Solve the system using elimination. Example: Solve the system using elimination. �3� � 2� � 5 �1� ��3� � 2� � �5 �1� �2� �2� 6� � 4� � 7 6� � 4� � 10 X _Y (Solution), 111 (Infinite), 000 (None) 3
MAT141, Chapter 2 Echelon Form S olving Applications Example: Some grocery store receipts from 1950 were found in a desk • Solving using elimination is essentially the technique drawer. One order listed three gallons of milk and two loaves of bread called the “echelon method” in the book. with a total of $2.70. A second order listed two gallons of milk and ▫ Rather than number the equations we refer to them as rows. three loaves of bread with a total price of $2.05. Determine the price ▫ Our goal is to “triangulate” a system, allowing us to work of each item. backwards to find solutions. S olving Applications (Infinite S olutions) S olving Applications (S upply & Demand) Example: A student production sells tickets at a cost of $5 Example: A company sells Star Wars collectible action figures to nerds that still live in their mother’s basements or have tolerant for students and $8 for non-students. They sold a total of wives. They have found after collecting data over the past 2 years that $68 worth of tickets. How many of each type did they sell? they have the following trends for monthly supply and demand. What is their equilibrium point? � � � � � 0.1105� � 100 � � � � � �0.0895� � 100 4
MAT141, Chapter 2 Parts of a Matrix • There is this guy named Morpheus and another named Neo, JK. • Matrices are a way of organizing data into rows and columns. • We can also take equations, organize them, and put them into matrices. • We like to separate the constants. In doing so we create Solutions of Linear Equations and Matrices what is called an augmented matrix. System Matrix Augmented Matrix Creating Augmented Matrices S olving using Gauss-Jordan Method Example: Turn each system into an augmented matrix. • The rules are very similar to elimination. • We need to be a little more systematic (a) ��2� � 3� � 10 about it though. � � � � 4 Carl Friedrich Gauss 1777-1855 (b) ��2� � � � �7 German 4� � 12 (c) �2� � � � 2 3� � 5 � 1 Wilhelm Jordan 1842-1899 German 5
MAT141, Chapter 2 Elimination and Gauss-Jordan Elimination and Gauss-Jordan Replace the second row with itself minus the first row. �� � Solution is � � �� and � � � �� S pecial Note Gauss-Jordan Method Example: Solve the system using Gauss-Jordan. • When combining two rows to replace one, you should not put a negative with the row you are replacing. �2� � 3� � 12 3� � 4� � 1 1 �1 3 1 �1 3 1 � 4�3� � 1 �1 3 2 � 2 � 4��1�� 6 � 4 1 4 � 4�1� � � � 4� � → 0 �11 This is OK 1 �1 3 1 �1 3 �1 � 4�3� � 1 �1 3 2 � �2 � 4��1�� �6� 4 1 �4 � 4�1� �� � � 4� � → 0 11 This is NOT OK It makes the term negative, which is actually ok for now but later this will cause a problem. 6
MAT141, Chapter 2 Gauss-Jordan Method Gauss-Jordan Method Example: Solve the system using Gauss-Jordan. Example: Solve the system using Gauss-Jordan. �3� � 2� � 5 ��3� � 2� � �5 6� � 4� � 7 6� � 4� � 10 Gauss-Jordan Method (On the TI) Gauss-Jordan Method (On the TI) Example: Solve the system using Gauss-Jordan. Example: A convenience store sells 23 sodas one summer afternoon in 12-, 16-, and 20-oz cups (small, medium, and large). The total � � � � 1 volume of soda sold was 376 oz. Suppose the price for a small, 2� � � � � � 10 � medium, and larger were $1, $1.25, and $1.40, respectively, with a 3� � 2� � 3� � �1 total amount of sales of $28.45. How many of each did the store sell? 1. 2 nd , Matrix ( � �� ) Edit, Enter 2. Fill in matrix 3. 2 nd , Quit (MODE) 4. 2 nd , Matrix ( � �� ) Math 5. rref( , enter 2 nd , Matrix ( � �� ), select your 6. matrix, enter 7. Enter Solution is � � 3, � � �2, � � 2 7
MAT141, Chapter 2 Gauss-Jordan Method (On the TI) To TI or not to TI? Example: Suppose the price for a small, medium, and large were $1, • I will ask you to perform the Gauss-Jordan method by $2, and $3, respectively, with a total amount of sales of $28.45. How hand on a system with 2 equations and two variables. many of each did the store sell? • For 3 or more, you only need to write the augmented matrix and your results obtained from the TI. Matrices • Recall that matrices are described by their rows and columns. Addition and Subtraction of Matrices • We say the size of a matrix is #���� � #������� 8
MAT141, Chapter 2 Matrices Matrices 2 1 0 and 2 0 are not equal. 3 1 1 and 3 0 0 1 are equal. �1 1 1 1 �2 3 1 0 1 � �2 � 1 0 � 1 � �3 3 � 0 3 0 � 1 �1 1 � �1 2 �1 2 1 1 0 2 � 1 1 � 0 3 1 0 � 1 � 0 � 1 � �1 �1 �1 � 1 �2 1 Matrices Matrices Example: Find the value of each variable. Example: Perform each operation. � � 0 � �3 3 1 0 � �3 0 3 (a) (a) �1 � 0 �1 2 0 �2 3 0 � � �3 � 7 8 � �2 3 1 (a) (a) 0 � 1 5 6 �3 1 �1 1 � � 2 �4 �2 0 1 � 8 �5 1 4 0 �2 0 1 (b) 0 1 0 � 0 4 1 � (b) 0 1 0 � 0 �4 �1 0 �3 �1 3 0 1 7 2 1 3 0 1 7 � � 1 1 �4 4 � 9
MAT141, Chapter 2 S calar Multiplication k � � � �� � �� � �� �� 3 �1 0 1 � �3 0 Multiplication of Matrices 2 6 3 Order of operations is also preserved. For example, 3 �1 0 1 � 2 �4 � �3 1 0 3 � 2 �4 � �1 1 1 2 1 6 1 7 �1 Labeling Parts of a Matrix Multiplying Matrices • We label each element of a matrix by its row and column. � �� � �� � �� � �� � �� � �� � �� � �� � �� � �� Example: The given matrices can be multiplied. Column 1 Row 2 1 �1 1 0 2 2 �1 2 ∙ 0 � �1 0 �1 1 �1 0 �2 2 � 2 2 � 3 2 � 3 Match 10
MAT141, Chapter 2 Multiplying Matrices Example: Multiply the given matrices. 0 �2 ∙ 4 (a) 1 1 2 • Let the result matrix guide you! � � �� � �� 1 2 1 ∙ 1 �1 � �� � �� 0 2 0 (b) 0 �2 1 0 2 ∙ • You get � �� by “multiplying” row 1 by column 2. 1 1 �1 1 0 � �� � 1 �1 � 2 0 � �1 Multiplying Matrices Book Practice • Note, the product of matrices is not commutative. • Let’s look at some more practice from the book. ▫ �� � �� • P. 83 (16, 20, 29, 44) • Show this with the following matrices. (16) �1 0 ∙ 6 5 ▫ � � 2 1 , � � 1 1 �1 7 2 0 1 0 6 0 �4 1 (20) 1 2 5 ∙ 2 10 �1 3 0 11
MAT141, Chapter 2 Book Practice Book Practice (29) 2 �2 4 3 7 0 (44) �1 ∙ 2 � 1 1 �1 5 Review: S olve Using Gauss-Jordan Review: S olve Using Gauss-Jordan � � � 2� � 5 ��2� � 5� � �26 3� � 2� � 12 � � 4� � 19 12
MAT141, Chapter 2 Review: S olve Using Gauss-Jordan Using Infinite S olutions � 2� � 6� � �8 • The solution from the previous problem was �5� � 15� � 20 ▫ �4 � 3�, � • What if x represented bowls and y represented plates. ▫ Your company can only make up to 10 plates in an hour. S olving Equations with Matrices • A system of equations. � � 3� � 2� � 4 � 2� � 7� � 3� � 8 3� � 8� � 5� � �4 • Turn this system of equations into a matrix equation. � 1 3 �2 4 � � 2 7 �3 8 Matrix Inverses � 3 8 �5 �4 Note that if you performed matrix multiplication you would get the equations from above. � � 3� � 2� � 4 13
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