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7.1 Solving Systems of Two Equations Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley What you’ll learn about The Method of Substitution Solving Systems Graphically The Method of Elimination Applications … and why Many applications in business and science can be modeled using systems of equations. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 2 Solution of a System A solution of a system of two equations in two variables is an ordered pair of real numbers that is a solution of each equation. A system is solved when all of its solutions are found. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 3 Example Using the Substitution Method Solve the system using the substitution method. 2 x y 10 6x 4 y 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Solving a Nonlinear System by Substitution Solve the system by substitution. x y 38 xy 361 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 5 Example Solving a Nonlinear System Algebraically Solve the system algebraically. y x 6x y 8x 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 6 Example Solving a Nonlinear System Graphically Solve the system: ln y x y x2 8x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 7 Example Using the Elimination Method Solve the system using the elimination method. 5 x 3 y 21 3x 2 y 5 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 8 Example Finding No Solution Solve the system: 3x 2 y 5 6 x 4 y 10 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 9 Example Finding Infinitely Many Solutions Solve the system. 3x 6 y 10 9 x 18 y 30 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 10 Example Solving Word Problems with Systems Find the dimensions of a rectangular cornfield with a perimeter of 220 yd and an area of 3000 yd2. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 11 Homework Homework Assignment #9 Read Section 7.2 Page 575, Exercises: 1 – 65 (EOO) Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 12 7.2 Matrix Algebra Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Quick Review The points (a) (1, 3) and (b) ( x, y) are reflected across the given line. Find the coordinates of the reflected points. 1. The x-axis 2. The line y x 3. The line y x Expand the expression, 4. sin( x y ) 5. cos( x y ) Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 14 Quick Review Solutions The points (a) (1, 3) and (b) ( x, y ) are reflected across the given line. Find the coordinates of the reflected points. 1. The x-axis (a) (1,3) (b) ( x, y ) 2. The line y x (a) ( 3,1) (b) ( y, x) 3. The line y x (a) ( 3, 1) (b) ( y, x) Expand the expression, 4. sin( x y ) sin x cos y sin y cos x 5. cos( x y ) cos x cos y sin x sin y Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 15 What you’ll learn about Matrices Matrix Addition and Subtraction Matrix Multiplication Identity and Inverse Matrices Determinant of a Square Matrix Applications … and why Matrix algebra provides a powerful technique to manipulate large data sets and solve the related problems that are modeled by the matrices. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 16 Matrix Let m and n be positive integers. An m × n matrix (read "m by n matrix") is a rectangular array of m rows and n columns of real numbers. a11 a 21 am1 a12 a22 am 2 a1n a2 n amn We also use the shorthand notation aij for this matrix. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 17 Matrix Vocabulary Each element, or entry, aij, of the matrix uses double subscript notation. The row subscript is the first subscript i, and the column subscript is j. The element aij is in the ith row and the jth column. In general, the order of an m × n matrix is m×n. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 18 Example Determining the Order of a Matrix What is the order of the following matrix? 1 4 5 3 5 6 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 19 Matrix Addition and Matrix Subtraction Let A aij and B bij both be matrices of order m n. 1. The sum A + B is the m n matrix A B aij bij . 2. The difference A B is the m n matrix A B aij bij . Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 20 Example Matrix Addition 1 2 3 2 3 4 4 5 6 5 6 7 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 21 Example Using Scalar Multiplication 1 2 3 3 4 5 6 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 22 The Zero Matrix The m n matrix 0 [0] consisting entirely of zeros is the zero matrix. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 23 Additive Inverse Let A aij be any m n matrix. The m n matrix B aij consisting of the additive inverses of the entries of A is the additive inverse of A because A B 0. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 24 Matrix Multiplication Let A aij be any m r matrix and B bij be any r n matrix. The product AB cij is the m n matrix where cij ai1b1 j +ai 2b2 j ... air brj . Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 25 Example Matrix Multiplication Find the product AB if possible. 1 2 3 A 0 1 1 1 0 and B 2 1 0 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 26 Identity Matrix The n n matrix I n with 1's on the main diagonal and 0's elsewhere is the identity matrix of order n n. 1 0 I n 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 27 Inverse of a Square Matrix Let A aij be an n n matrix. If there is a matrix B such that AB BA I n , then B is the inverse of A. We write B A1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 28 Inverse of a 2 × 2 Matrix 1 a b 1 d b If ad bc 0, then . ad bc c a c d The number ad bc is the determinant of the 2 2 matrix a b a b A and is denoted det A ad cb. c d c d Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 29 Minors and Cofactors of an n × n Matrix If A is an n n matrix where n 2, the minor M ij corresponding to the element aij is the determinant of the n 1 n 1 matrix obtained by deleting the row and column containing aij . The cofactor corresponding to aij is Aij 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley i j M ij . Slide 7- 30 Determinant of a Square Matrix Let A aij be a matrix of order n n (n 2). The determinant of A, denoted by det A or | A | , is the sum of the entries in any row or any column multiplied by their respective cofactors. For example, expanding by the i th row gives det A | A | ai1 Ai1 ai 2 Ai 2 ... ain Ain . Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 31 Transpose of a Matrix Let A aij be a matrix of order n m. The transpose of A, denoted by AT is the matrix in which the rows in A become the columns in AT and the columns in A become the rows in AT or AT a ji . Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 32 Example Using the Transpose of a Matrix If pizza sizes are given by the matrix Size Pers Sm Med Larg , pizza sales are given by the matrix Sales 55 25 15 10 , and pizza prices are given by the matrix Price $2.50 $3.50 $7.50 $11.50 , what are the total sales for the day? Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 33 Inverses of n × n Matrices An n × n matrix A has an inverse if and only if det A ≠ 0. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 34 Example Finding Inverse Matrices 1 3 Find the inverse matrix if possible. A 2 5 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 35 Properties of Matrices Let A, B, and C be matrices whose orders are such that the following sums, differences, and products are defined. 1. Community property Addition: A + B = B + A Multiplication: Does not hold in general 2. Associative property Addition: (A + B) + C = A + (B + C) Multiplication: (AB)C = A(BC) 3. Identity property Addition: A + 0 = A Multiplication: A·In = In·A = A 4. Inverse property Addition: A + (-A) = 0 Multiplication: AA-1 = A-1A = In |A|≠0 5. Distributive property Multiplication over addition: A(B + C) = AB + AC (A + B)C = AC + BC Multiplication over subtraction: A(B - C) = AB - AC (A - B)C = AC - BC Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 36 Reflecting Points About a Coordinate Axis To reflect a point about the x-axis, express the point as a 1 0 1 2 matrix and multiply by to obtain the 1 2 0 1 matrix of the reflected point. To reflect a point about the y -axis, express the point as a 1 0 1 2 matrix and multiply by to obtain the 1 2 0 1 matrix of the reflected point. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 37 Example Using Matrix Multiplication 46. A company has two factories, each manufacturing three products. The number of products i made in factory j in one week is given by aij in the matrix 120 70 A 150 110 . If production is increased by 10%, write the new production levels 80 160 as a matrix B. How is B related to A? Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 38