1. Chapter 10: Systems of Equations and Inequalities 10.1 Systems of Linear Equations; Substitutionor Elimination (Stack/Add) Chapter 10: Systems of Equations and Inequalities 10.1 Systems of Linear Equations; Substitutionor Elimination (Stack/Add)

2. 1. Definitions An equation is linear if it can be written as: System of linear equations. Collection of 2 or more equations containing one or more variables. Solution to a system of equations. The values of the variables which make all the equations true. Definition Example

3. 1. More Definitions Consistent Consistent – System with at least one solution Two types of Consistent Solutions: Dependent- System with infinitely many solutions Independent – System with only one solution Inconsistent Inconsistent – System with no solutions Definition

4. 2. Verify a solution Verify that (4,-1) is a solution to: Is (-4,3) a solution ?

5. 3. Methods for Solving System of Linear Equations 1)Substitution 2)Elimination (Stack/Add) 3)Graphing (For system of 2 variables) 4)Section 10.2: Matrices What does the graph of this equation look like?

6. 4. Method of Substitution Goal: Convert to equation of one variable. Verify Solution when finished!

7. 5. Method of Elimination Goal Goal: Add 2 equations together to eliminate a variable Use Use: When a variable cannot be easily isolated #1

8. 5. Method of Elimination 5. Method of Elimination (Stack/Add) Verify Solution! #2 1) Equations should be of form: 1) Equations should be of form: Ax + By = C and variables lined up 2) Multiply by nonzero number so a variable cancels when adding 3) Add the equations 4) Solve the new equation 5) Back-substitute 1) Equations should be of form: 1) Equations should be of form: Ax + By = C and variables lined up 2) Multiply by nonzero number so a variable cancels when adding 3) Add the equations 4) Solve the new equation 5) Back-substitute

9. 6. Inconsistent System What is the graph of this system? #3 There is no solution when the result is a false statement with no variables involved Examples : 0 = 2 -1 = 5 There is no solution when the result is a false statement with no variables involved Examples : 0 = 2 -1 = 5

10. 7. Dependent system The solution of a Dependent System is a set: {(x,y) | } #4 Infinitely many solutions if your solution results in a statement that is always true. Examples: 2 = 2 0 = 0 -3/4 = -3/4 Infinitely many solutions if your solution results in a statement that is always true. Examples: 2 = 2 0 = 0 -3/4 = -3/4

11. 8. Applications Solving an application problem: Step 1: Define the variables Step 2: Write the system in words (describe verbally) Step 3: Plug in variables for the words. Step 4: Solve the system. p. 739 #58, 62

12. 9. System of 3 Linear Equations GOAL GOAL: Reduce the system to 2 equations in the same 2 variables and solve for the 2 variables.

13. 10. Example of Dependent 3x3 Solve: What does solution look like on graph? Write Solution as {(x,y) | }

14. Polya’s 4 Principles for solving word problems Polya’s 4 Principles for solving word problems 1.Understand the problem 1.Understand the problem: Read carefully (annotate, highlight) What are you asked to find or show ? (variables) Can you restate the problem in your own words? Can you think of a picture/diagram/table to help you understand the problem? Do you understand all the words used in stating the problem? Do you need to ask a question to get the answer? ( Hint: “I don’t understand” is not a question!) 2.Devise a plan Make a list, look for a pattern, work backward, be creative 3.Carry out the plan Persistence and patience pay off. 4.Review/extend 4.Review/extend: Reflect on what worked and what didn’t to predict the strategy to use in future problems.

15. Traffic Control: I2I2 I1I1 I3I3