1. Copyright ©2015 Pearson Education, Inc. All rights reserved.

2. Chapter 6  Systems of Linear Equations and Matrices Copyright ©2015 Pearson Education, Inc. All rights reserved.

3. Section 6.1  Systems of Two Linear Equations in Two Variables Copyright ©2015 Pearson Education, Inc. All rights reserved.

4. Slide 1 - 4 Solve the system Solution: Copyright ©2015 Pearson Education, Inc. All rights reserved. Example: The multipliers 2 and −3 were chosen so that the coefficients of x in the two equations would be negatives of each other. Any solution of both of these equations must also be a solution of their sum: Multiply the first equation by 2 and the second equation by −3 to get To find the corresponding value of x, substitute 2 for y in either of the original equations.

5. Slide 1 - 5 Solve the system Solution: Copyright ©2015 Pearson Education, Inc. All rights reserved. Example: We choose the first equation: Therefore, the solution of the system is The graphs of both equations of the system are shown below. They intersect at the point the solution of the system.

6. Section 6.2  Larger Systems of Linear Equations Copyright ©2015 Pearson Education, Inc. All rights reserved.

7. Slide 1 - 7 Copyright ©2015 Pearson Education, Inc. All rights reserved.

8. Slide 1 - 8 Copyright ©2015 Pearson Education, Inc. All rights reserved.

9. Slide 1 - 9 Use matrices to solve the system Solution: Copyright ©2015 Pearson Education, Inc. All rights reserved. Example: Write the augmented matrix and perform row operations to obtain a first column whose entries (from top to bottom) are 1, 0, 0: Stop! The second row of the matrix denotes the equation Since the left side of this equation is always zero and the right side is 2, it has no solution. Therefore, the original system has no solution.

10. Section 6.3  Applications of Systems of Linear Equations Copyright ©2015 Pearson Education, Inc. All rights reserved.

11. Slide 1 - 11 Solution: Copyright ©2015 Pearson Education, Inc. All rights reserved. Example: Let correspond to the year 2000. Then the table represents data points (22, 334), (40, 380), and (50, 400). We must find a function of the form whose graph contains these three points. If (20, 334) is to be on the graph, we must have that is, The table shows Census Bureau projections for the population of the United States (in millions). Use the given data to construct a quadratic function that give the U.S. population (in millions) in year x.

12. Slide 1 - 12 Solution: Copyright ©2015 Pearson Education, Inc. All rights reserved. Example: The other two points lead to these equations: Now work by hand or use technology to solve the following system: The table shows Census Bureau projections for the population of the United States (in millions). Use the given data to construct a quadratic function that give the U.S. population (in millions) in year x.

13. Slide 1 - 13 Solution: Copyright ©2015 Pearson Education, Inc. All rights reserved. Example: The reduced row echelon form of the augmented matrix shown in the figure below shows that the solution is So the function is The table shows Census Bureau projections for the population of the United States (in millions). Use the given data to construct a quadratic function that give the U.S. population (in millions) in year x.

14. Section 6.4  Basic Matrix Operations Copyright ©2015 Pearson Education, Inc. All rights reserved.

15. Slide 1 - 15 Copyright ©2015 Pearson Education, Inc. All rights reserved.

16. Slide 1 - 16 Find each sum if possible. Solution: Copyright ©2015 Pearson Education, Inc. All rights reserved. Example: (a) Solution: Because the matrices are the same size, they can be added by finding the sum of each corresponding element. (b) The matrices are of different sizes, so it is not possible to find the sum

17. Slide 1 - 17 Copyright ©2015 Pearson Education, Inc. All rights reserved.

18. Slide 1 - 18 Copyright ©2015 Pearson Education, Inc. All rights reserved.

19. Slide 1 - 19 Copyright ©2015 Pearson Education, Inc. All rights reserved.

20. Slide 1 - 20 Copyright ©2015 Pearson Education, Inc. All rights reserved.

21. Section 6.5  Matrix Products and Inverses Copyright ©2015 Pearson Education, Inc. All rights reserved.

22. Slide 1 - 22 Copyright ©2015 Pearson Education, Inc. All rights reserved.

23. Slide 1 - 23 Copyright ©2015 Pearson Education, Inc. All rights reserved.

24. Slide 1 - 24 Copyright ©2015 Pearson Education, Inc. All rights reserved.

25. Slide 1 - 25 Given matrices A and B as follows, determine whether B is the inverse of A: Solution: Copyright ©2015 Pearson Education, Inc. All rights reserved. Example: B is the inverse of A if so we find those products. Therefore, B is the inverse of A; that is, (It is also true that A is the inverse of B, or

26. Slide 1 - 26 Copyright ©2015 Pearson Education, Inc. All rights reserved.

27. Section 6.6  Applications of Matrices Copyright ©2015 Pearson Education, Inc. All rights reserved.

28. Slide 1 - 28 Copyright ©2015 Pearson Education, Inc. All rights reserved.

29. Slide 1 - 29 The next step cannot be performed because of the zero in the second row, second column. Use the inverse of the coefficient matrix to solve the system Solution: Copyright ©2015 Pearson Education, Inc. All rights reserved. Example: The coefficient matrix is A graphing calculator will indicate that does not exist. If we carry out the row operations, we see why: Therefore, the inverse does not exist, and the system has no solutions.