System of Linear Equation in Two Variables(COT).pptx

Presentation about system of linear equation in two variables

1. System of Linear Equation in Two Variables Math 8

2. Learning Competency (MELC) Illustrates a system of linear equations in two variables Graph a system of linear equations in two variables Categorizes when a given system of linear equations in two variables has graphs that are parallel, intersecting, and coinciding. Specific Learning Objectives: To define a system of linear equations in two variables To identify the three types of system of linear equations in two variables; and To describe the graph of the systems of linear equations in two variables as parallel, intersecting, or coinciding; To graph systems of linear equations in two variables. To represent real-life situations using systems of linear equations in two variables.

3. System of Linear Equations in Two Variables A system of linear equations consists of two or more linear equations. It can have no solution, one solution or infinite solutions.   “The system of equations and 𝑥 − 𝑦 = 1”  

4. Solutions of Linear Equations in Two Variables            

5. Graphing the System of Linear Equation Use slope and y intercept to graph the system of linear equation     m = - 1 y - int : (0, 5)   m = 1 y - int : (0, -1)   Therefore, (3, 2) satisfies both the equation Note: You can use any method in graphing.

6. Types of Systems of Linear Equations in Two Variables

7. 3 Types of Systems of Linear Equations in Two Variables 1. Consistent and Independent 2. Consistent and Dependent 3. Inconsistent

8. 3 Types of Systems of Linear Equations in Two Variables

9.   3 Types of Systems of Linear Equations in Two Variables     Consistent and Independent       Since, Slope and y – intercept are both different then: The graph is Intersecting

10. TRY THIS!   1.   2.   3.

11. TRY THIS!   1.         m = 1 y - int : (0, -4) m = -1 y - int : (0, 2)   Slope and y – intercept are both different then: Consistent and Independent The graph is Intersecting. It means one solution exist

12. TRY THIS!   2.       m =   y - int : (0, 2) m =   y - int : (0, 1)   The same slope but different y - intercept Inconsistent The graph is Parallel. It means no solution exists.

13. TRY THIS!   3.       m =   y - int : (0, -3) m =   y - int : (0, -3)   The same slope and y – intercept, so: Consistent and Dependent The graph are Coinciding. It has infinite solution .

14. Example in Real World A mobile network provider offers a postpaid sim-only plan that costs Php999 per month plus Php2.50 per text message sent to other networks. Another mobile network sim-only plan costs Php1299 per month but offers Php1 only for every text message sent to other networks. What are the two equations that can be used to represent the total monthly cost (𝑦) of the number of text messages (𝑥) sent to other networks?

15. Example in Real World A mobile network provider offers a postpaid sim-only plan that costs Php999 per month plus Php2.50 per text message sent to other networks. Another mobile network sim-only plan costs Php1299 per month but offers Php1 only for every text message sent to other networks. What are the two equations that can be used to represent the total monthly cost (𝑦) of the number of text messages (𝑥) sent to other networks? Solution: Let 𝑥 - be the total number of text messages sent to other networks 𝑦 - be the total monthly cost of x text messages sent to other networks    

16. Solving System of Linear Equation in Two Variables Math 8

17. Learning Competency (MELC) Solves a system of linear equations in two variables by a. Graphing b. Substitution; c. Elimination. Specific Learning Objectives: solve systems of linear equations in two variables by substitution and elimination methods; solve problems involving systems of linear equations in two variables by graphing, substitution and elimination; and determine the most efficient method in solving system of linear equations in two variables.

18. Solving System of Linear Equations by Substitution

19. The Substitution Method You solve one equation for one of the variables, then substitute the new form of the equation into the other equation for the solved variable.

20. Example 1   1. Solve the following system using the substitution method equation 1 equation 2 STEP 1: Solve one of the equations for a variable.       (subtract 3x from both sides) (multiply both sides by – 1) Continued. equation 3

21. Example 1 STEP 2: Substitute the expression from step 1 into the other equation and solve the new equation                 (replace y with result from first equation) (use the distributive property) (add 12 to both sides) (divide both sides by 2)) Continued.

22. Example 1 STEP 3: Substitute the value found in step 2 into either equation containing both variables.             Continued. equation 3 Solution Set

23. Example 1 STEP 4: Check the proposed solution in the original equations.   Continued. Solution Set     (eq 1) (eq 2)            

24. Solving a System of Linear Equations by the Substitution Method STEP 4: Check the proposed solution in the original equations. STEP 1: Solve one of the equations for a variable. STEP 2: Substitute the expression from step 1 into the other equation and solve the new equation STEP 3: Substitute the value found in step 2 into either equation containing both variables.

25. Solving System of Linear Equations by Elimination

26. The Substitution Method Another method that can be used to solve systems of equations is called the addition, arithmetic or elimination method . You multiply both equations by numbers that will allow you to combine the two equations and eliminate one of the variables.

27. Example 1   1. Solve the following system using the substitution method equation 1 equation 2 STEP 1: Rewrite each equation in standard form, eliminating fraction coefficients     Continued. DONE STEP 2: If necessary, multiply one or both equations by a number so that the coefficients of a chosen variable are opposites.  

28. Example 1 STEP 3: Add the equations.           (divide both sides by 2) STEP 4: Find the value of one variable by solving equation from step 3.  

29. Example 1 STEP 5: Find the value of the second variable by substituting the value found in step 4 into either original equation. Let             Continued. Solution Set      

30. Example 1 STEP 6: Check the proposed solution in the original equations   Solution Set     (eq 1) (eq 2)            

31. Example in Real World The sum of Janna age and Mark’s age is 40. Two years ago, Janna was twice as old as Mark. Find Janna’s age now.

32. Example in Real World Step 1: Understand the problem. The sum of Janna age and Mark’s age is 40. Two years ago, Janna was twice as old as Mark. Find Janna’s age now. Let x = Janna’s age Let y = Mark’s age

33. Example in Real World Step 2: Devise a plan (translate). The sum of Janna age and Mark’s age is 40. Two years ago, Janna was twice as old as Mark. Find Janna’s age now. x + y = 40 For equation 1: The sum of Janna and Mark's age is 40 For equation 2: Two years ago, Janna was twice as old as Mark. Janna (two years ago) = 2 times Mark’s age (two years ago) 𝑥 − 2 = 2(𝑦 − 2)

34. Example in Real World For equation 2: Two years ago, Janna was twice as old as Mark. Janna (two years ago) = 2 times Mark’s age (two years ago) 𝑥 − 2 = 2(𝑦 − 2) 𝑥 − 2 = 2𝑦 − 4 𝑥 − 2y = -2 x + y = 40 𝑥 − 2y = -2 equation 1 equation 2

35. Example in Real World Step 3: Carry out the plan (solve) Use substitution method: x + y = 40 𝑥 − 2y = -2 equation 1 equation 2 x + y = 40 (eq1) y = -x + 40 (eq3) 𝑥 − 2y = -2 (eq 2) 𝑥 − 2(-x+ 40) = -2 Substitute the new equation formed in eq3 𝑥 + 2x - 80 = -2 3x = -2 + 80 3x = 78 Divide both side by 3 to get the value of x x = 26 New equation formed from the equation 1 To find Janna’s age

36. Example in Real World x + y = 40 Let x = 26 Use equation 1 to find the value of y 26 + y = 40 y = 40 - 26 y = 14 To find Mark’s age

37. Example in Real World Step 4: Look back and interpret Check answers directly against the facts of the problems. Substitute the value of 𝑥 and 𝑦 to both equations Sum of Janna’s and Mark’s age x + y = 40 26 + 14 = 40 40 = 40 Two years ago, Janna was twice as old as Mark x − 2y = -2 26 − 2(14) = -2 26 − 28 = -2 -2 = -2 Therefore, Janna’s age now is 26

38. Thank you! MA. RICA MAE M. ROSETE Position Author / Learning Resource Developer

39. For Image Resources https://www.vecteezy.com/free-vector/teacher

40. Refences Ulpina , J.N.(2014). Math Builders 8. JO-ES Publishing House DepEd CO Modules