1. 1 What you will learn  We need to review several concepts from Algebra II: Solving a system of equations graphically Solving a system of equations algebraically Matrix operations Matrix transformations Solving a system of equations using matrices Graphing linear inequalities

2. Objective: Brain Dump of Chapter 2 2 Some Vocabulary A system of two linear equations in two variables, x and y consists of two equation of the following form: Ax + By = C Dx + Ey = F A solution of a system of linear equations is an ordered pair (x, y) that satisfies each equation.

3. Objective: Brain Dump of Chapter 2 3 How Do You Solve These Things? The first method we will use is to graph the two equations and see where they intersect. Example: Solve the system 2x – 3y = 1 x + y = 3

4. Objective: Brain Dump of Chapter 2 4 Number of Solutions Infinite NumberOne SolutionNo Solution Solutions

5. Objective: Brain Dump of Chapter 2 5 You Try!  Use substitution to find the solution to the following: 3x – y = 13 2x + 2y = -10

6. Objective: Brain Dump of Chapter 2 6 You Try!  Solve using the linear combination method: 2x + 3y = 5 x – 5y = 9

7. Objective: Brain Dump of Chapter 2 7 Many or No Solutions  When you solve a system of equations and you get something that is impossible (e.g. 6 = 7) then the system has no solution.  When you solve a system of equation and you get something that is always true (0 = 0), then the system has an infinite number of solutions.

8. Objective: Brain Dump of Chapter 2 8 Matrices 2 rows 3 columns entry

9. Objective: Brain Dump of Chapter 2 9 Adding and Subtracting Matrices You can add or subtract matrices only if they have the same dimensions!! To add or subtract you simply add or subtract the corresponding entries. A.B.C.

10. Objective: Brain Dump of Chapter 2 10 Scalar Multiplication In matrix algebra, a real number is often called a scalar. To multiply a matrix by a scalar, you multiply each entry in the matrix by the scalar. This process is called scalar multiplication. A.B.

11. Objective: Brain Dump of Chapter 2 11 Multiplying Matrices…This isn’t for the fearful In order to multiply two matrices A and B, the number of columns in A must be equal to the number of columns in B. Example: The answer matrix will be… Question? Could I multiply matrix B times matrix A?

12. Objective: Brain Dump of Chapter 2 12 More Fun With Matrices A = B =C = 1. Find AB2. Find BA 3. Find A(B + C)

13. Objective: Brain Dump of Chapter 2 13 Transformations with Matrices  You can dilate, translate, reflect and rotate figures using matrices.  Dilate a triangle with vertices at X(0,8), Y(5,9), and Z (-3,2) by a scale factor of 2.

14. Objective: Brain Dump of Chapter 2 14 Translating Translate quadrilateral ABCD 2 units to the left and 4 units up. A(-1, 1) B(4, 0) C(4, -5),and D(-1, -3)

15. Objective: Brain Dump of Chapter 2 15 Reflecting Using Matrices  Figures can be reflected over the x axis, y axis or over the line y = x. Use a reflection matrix and matrix multiplication.  Square ABCD has vertices at (-1,2), (-4,1), (-3,-2) and D(0,-1). Find the image of the square after a reflection over the y-axis:

16. Objective: Brain Dump of Chapter 2 16 List of Transformation Matrices  Page 90 and 91 in your book.  Remember, you put the transformation matrix first and then multiply it by the matrix with the x and y coordinates of the figures.

17. Objective: Brain Dump of Chapter 2 17 How Do You Get This Thing? Determinant of a 2 x 2 matrix: = ad - cb

18. Objective: Brain Dump of Chapter 2 18 Finding the Determinant for a 3 x 3 =(aei+bfg+cdh)–(gec+hfa+idb)

19. Objective: Brain Dump of Chapter 2 19 Example  Find the inverse of A =

20. Objective: Brain Dump of Chapter 2 20 We Have Arrived Solving a system of equations using matrices involves setting up three matrices. The first matrix is a matrix that includes the coefficients (numbers “attached” to x and y). The second matrix contains the variables and acts only as a placeholder. The third matrix contains the constants, or “answer” side of the equations.

21. Objective: Brain Dump of Chapter 2 21 Setting Up the Matrices  Write the system of linear equations as a matrix equation: -3x + 4y = 5 2x – y = -10

22. Objective: Brain Dump of Chapter 2 22 What Next  Remember: AX = B If we multiply both sides of the equation by the inverse of matrix A we can “solve” the equation.

23. Objective: Brain Dump of Chapter 2 23 Another Example  Use a matrix equation to solve the following 2x + 3y + z = -1 3x + 3y + z = 1 2x + 4y + z = -2

24. Objective: Brain Dump of Chapter 2 24 Homework  Start working on the Chapter 2 Review packet. It will be due at the end of class Friday.  You will have a chance to do some in-class work on Wednesday.  We will do corrections and go over tricky problems on Friday.  We will have a practice “quest” on Monday.  You will have a quest on this material next Wednesday.