1. Systems of Equations

2. I. Systems of Linear Equations Four Methods: 1. Elimination by Substitution 2. Elimination by Addition 3. Matrix Method 4. Cramer’s Rule 5. Geometric Method for a 2 by 2 system

3. Example (1)

4. Geometric method Graph the lines represented by the equations The solution is the intersection of these lines

5. Example (2)

6. Geometric method Graph the lines represented by the equations These lines are parallel and do not intersect, and so no solution for the given system exists.

7. Example (3)

8. Geometric method Graph the lines represented by the two equations (they are equivalent equations) representing the same lines

9. 3 by 3 Linear system See the following examples: 1. Example (5) Page 144 2. Example (6) Page 145 3. Example (7) Page 146

10. Cramer’s Rule

11. Determinants

12. Two by Two Determinants

13. Systems of Linear Equations

14. Two Equations in Two Unknowns

15. Example

16. The case when Δ 0 = 0 The left side of the first equation is a k multiple of the left side of the second one, for some real number k The right side of the first equation is a k multiple of the right side of the second one → There are finitely many solutions for the system The right side of the first equation is not a k multiple of the right side of the second one. → There is no solution for the system

19. Three by Three Determinants

20. A System of Three Linear Equations in Three Unknowns

21. Example

22. Nonlinear System Example (1)

24. Geometric method Graph the line and the quadratic function represented by the two equations; y = - x 2 + 2x + 7 and y = 3x + 1 The points of their intersection are the solutions of the system

25. Intersection of Graphs

26. Example (1)

29. Why we get two answers, when actually the graphs intersect at only one point?

30. Answer: Because, when we squared √( x+2), we introduced the other function whose square is also equal x+2 Which function is this? At which point does it intersect the line y = x+2

31. It is the function y = - √( x+2) It intersects the line y = x+2 at the point whose x coordinate is 7. What is the y-coordinate of this point?