Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 7.1 Solving Systems of Two Equations
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 2 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- 3 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- 4 Example Using the Substitution Method
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 5 Example Solving a Nonlinear System by Substitution
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 6 Example Solving a Nonlinear System Algebraically
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 7 Example Solving a Nonlinear System Graphically
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 8 Example Using the Elimination Method
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 9 Example Finding No Solution
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding Infinitely Many Solutions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Solving Word Problems with Systems Find the dimensions of a rectangular cornfield with a perimeter of 220 yd and an area of 3000 yd 2.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework Homework Assignment #9 Read Section 7.2 Page 575, Exercises: 1 – 65 (EOO)
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 7.2 Matrix Algebra
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Quick Review
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Quick Review Solutions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 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 Matrix
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Matrix Vocabulary Each element, or entry, a ij, of the matrix uses double subscript notation. The row subscript is the first subscript i, and the column subscript is j. The element a ij is in the i th row and the j th column. In general, the order of an m × n matrix is m×n.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Determining the Order of a Matrix
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Matrix Addition and Matrix Subtraction
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Matrix Addition
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Using Scalar Multiplication
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide The Zero Matrix
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Additive Inverse
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Matrix Multiplication
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Matrix Multiplication
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Identity Matrix
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Inverse of a Square Matrix
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Inverse of a 2 × 2 Matrix
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Minors and Cofactors of an n × n Matrix
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Determinant of a Square Matrix
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Transpose of a Matrix
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Using the Transpose of a Matrix
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 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 Example Finding Inverse Matrices
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 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·I n = I n ·A = A 4. Inverse property Addition: A + (-A) = 0 Multiplication: AA -1 = A -1 A = I n |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 Reflecting Points About a Coordinate Axis
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Using Matrix Multiplication