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7.1

7.1. Solving Systems of Two Equations. 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. Solution of a System.

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7.1

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  1. 7.1 Solving Systems of Two Equations

  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.

  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.

  4. Example Using the Substitution Method

  5. Example Solving a Nonlinear System by Substitution

  6. Example Solving a Nonlinear System Algebraically

  7. Example Solving a Nonlinear System Graphically

  8. Example Using the Elimination Method

  9. Example Finding No Solution

  10. Example Finding Infinitely Many Solutions

  11. 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.

  12. Homework • Homework Assignment #9 • Read Section 7.2 • Page 575, Exercises: 1 – 65 (EOO)

  13. 7.2 Matrix Algebra

  14. Quick Review

  15. Quick Review Solutions

  16. 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.

  17. Matrix

  18. 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 orderof anm × n matrix is m×n.

  19. Example Determining the Order of a Matrix

  20. Matrix Addition and Matrix Subtraction

  21. Example Matrix Addition

  22. Example Using Scalar Multiplication

  23. The Zero Matrix

  24. Additive Inverse

  25. Matrix Multiplication

  26. Example Matrix Multiplication

  27. Identity Matrix

  28. Inverse of a Square Matrix

  29. Inverse of a 2 × 2 Matrix

  30. Minors and Cofactors of an n × n Matrix

  31. Determinant of a Square Matrix

  32. Transpose of a Matrix

  33. Example Using the Transpose of a Matrix

  34. Inverses of n× n Matrices An n× n matrix A has an inverse if and only if det A≠ 0.

  35. Example Finding Inverse Matrices

  36. 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

  37. Reflecting Points About a Coordinate Axis

  38. Example Using Matrix Multiplication

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