Illustrate Linear Equation in two variables.pptx

Illustrate Linear Equation in two variables and differentiate linear inequality and equality in two variable for mathematics 8 in quarter 2. Illustrate Linear Equation in two variables and differentiate linear inequality and equality in two variable for mathematics 8 in quarter 2. Illustrate Linear Equation in two variables and differentiate linear inequality and equality in two variable for mathematics 8 in quarter 2. Illustrate Linear Equation in two variables and differentiate linear inequality and equality in two variable for mathematics 8 in quarter 2. Illustrate Linear Equation in two variables and differentiate linear inequality and equality in two variable for mathematics 8 in quarter 2. Illustrate Linear Equation in two variables and differentiate linear inequality and equality in two variable for mathematics 8 in quarter 2. Illustrate Linear Equation in two variables and differentiate linear inequality and equality in two variable for mathematics 8 in quarter 2. Illustrate Linear Equation in two variables and differentiate linear inequality and equality in two variable for mathematics 8 in quarter 2. Illustrate Linear Equation in two variables and differentiate linear inequality and equality in two variable for mathematics 8 in quarter 2. Illustrate Linear Equation in two variables and differentiate linear inequality and equality in two variable for mathematics 8 in quarter 2. Illustrate Linear Equation in two variables and differentiate linear inequality and equality in two variable for mathematics 8 in quarter 2. Illustrate Linear Equation in two variables and differentiate linear inequality and equality in two variable for mathematics 8 in quarter 2. Illustrate Linear Equation in two variables and differentiate linear inequality and equality in two variable for mathematics 8 in quarter 2. Illustrate Linear Equation in two variables and differentiate linear inequality and equality in two variable for mathematics 8 in quarter 2. Illustrate Linear Equation in two variables and differentiate linear inequality and equality in two variable for mathematics 8 in quarter 2. Illustrate Linear Equation in two variables and differentiate linear inequality and equality in two variable for mathematics 8 in quarter 2. Illustrate Linear Equation in two variables and differentiate linear inequality and equality in two variable for mathematics 8 in quarter 2. Illustrate Linear Equation in two variables and differentiate linear inequality and equality in two variable for mathematics 8 in quarter 2. Illustrate Linear Equation in two variables and differentiate linear inequality and equality in two variable for mathematics 8 in quarter 2. Illustrate Linear Equation in two variables and differentiate linear inequality and equality in two variable for mathematics 8 in quarter 2. Illustrate Linear Equation in two variables and differentiate linear inequality and equality in two variable for mathematics 8 in quarter 2. Illustrate Linear Equation in two varia

1. ILLUSTRATE LINEAR EQUATION IN TWO VARIABLES

2. What are Linear Equations in Two Variables? Linear equations in two variables is an equation of the form ax + by + c = 0, where x and y are the two variables and a, b, and c are real numbers and a and b are non-zero. It is popularly known as simultaneous linear equation. Linear equations in two variables are usually used in geometry to find the coordinates of a straight line. Example: x+y –3=0 is a linear equation in two variables x and y

3. Solutions of Linear Equations in Two Variables The solution of a linear equation ax+by =c, where a, b and c are constants are the set of ordered pair ( x,y ) that will satisfy the equation or that will make the equation true. To find the solution of a linear equation in two variables, we usually express one variable in terms of the other variable

4. EXAMPLE Determine whether the ordered pair ( -3 , 5 ) is a solution of y = 4x – 3 Solution: y = 4 x – 3 5 = 4( –3 ) – 3 Replace x with –3 and y with 5. 5 = –12 – 3 5  –15 Because the equation is not true, (–3, 5) is not a solution for y = 4 x – 3.

5. EXAMPLE Determine whether the ordered pair ( 2 , 3 ) is a solution of x + 2y = 8 Solution: x + 2y = 8 2 + 2( 3 ) = 8 2 + 6 = 8 8 = 8

6. Check the ordered pair that is a solution to the given linear equation.   1.) (9,1) (-9,1)   2.) (5,2) (2,5)   3.) (2,4) (-2,4) 4.) (4,2) (-4,2) 5.) (0,-5) (5,0)  

7. Check the ordered pair that is a solution to the given linear equation.   1.) (9,1) (-9,1)   2.) (5,2) (2,5)   3.) (2,4) (-2,4) 4.) (4,2) (-4,2) 5.) (0,-5) (5,0)  

8. “TRY ME” Determine if the given ordered pair is the solution of linear equation in two variables. ( SOLUTION or NOT ) 1.) 5x – y = -1 (1,6) 2.) 4x + 5y = 10 (2,4) 3.) y = 3x + 2 (2,8) 4.) y = 4x + 1 (4,17)

9. INDIVIDUAL ACTIVITY Determine the solution of linear equation in two variables. 1.) y = 4x + 1 2.) y = -2x + 6 3.) 3x + y = 9 x -2 -1 y x 2 4 y x 1 3 y

10. Linear Inequalities in Two Variables vs Linear Equations in Two Variables.

11. Objectives Identify the solution of a linear equation or inequality in two variables. b. Determine whether a point is a solution of a linear inequality or not. c. Appreciate the concept of linear inequality in two variables.

12. REVIEW Which of the following points is a solution to the following linear inequalities? Explain your answer. 1. 2x – y > - 3 (3, 6) (4, 11) (2, 7) 2. y ≥ -6x + 1 (2, -11) (-3, -8) (-5, 6) 3. 5x + y > 10 (0, -3) (3, -5) (4, 8) 4. y ≤ x - 9 (2, -5) (9, -3) (12, 3) 5. y < 5x -3 (3, 5) (2, 7) (3, 0 )

13. EXAMPLES The points (2, 7), (0, 3) and (-1, 1) are the points on the line y = 2x + 3 and the solutions to the given equation.   Determine if the points (5, 8), (0, 0), and (10, 10) are solutions to the linear inequality 2x + 5y > 10. Linear EQUALITY in two variable Linear INEQUALITY in two variable

14. 1. What can you say about the solution of a linear equation? 2. When can you say that a point is a solution to a linear inequality in two variables? 3. How can you solve if a point is a solution to a linear equation or inequality in two variables?

15. Fill in the blanks then state whether each given ordered pair is a solution of the inequality. 1. x + 2y ≤ 8; (6,1) x = ___ and y = ___ ___ + 2 (___) ≤ 8 6 + ___ ≤ 8 ___ ≤ 8 ________ (True or False) Thus, _________________________ (write your conclusion)

16. 2. x - ≥ -2: (-6, -8) x = ___ and y = ___ ___ - ( ___)≥-2 -6 + ___≥-2 ___≥-2 ________ (True or False) Thus, _________________________ (write your conclusion)

17. 3. 2x – y 7: (3, -1) x = ___ and y = ____ 2(___)- ___< 7 ___ + ___< 7 ___< 7 ________ (True or False) Thus, _________________________ (write your conclusion)

18. 4. 3x – y > 6;(0,0) x = ___ and y = ____ 3(___) + ___>6 ____ + ___>6 ________ (True or False) Thus, _________________________ (write your conclusion)

19. 5. x + y ≤ 8; (5,4) x = ___ and y = ___ ___ +___ ≤ 8 ___ ≤ 8 ________ (True or False) Thus, _________________________ (write your conclusion)

20. Connect the following coordinates to the linear inequality that makes them a solution. Show your solution. 1. (8,2) • 2. (-1, 2) • • 2x – y > 5 3. (0, 5) • 4. (0,0) • • x + 2y ≤ 1 5. (2, 5) • DEVELOPING MASTERY

21. Determine 2 solution for each of the following linear inequalities. Show your solution. 1. 5x + 2y < 17 ; x = 3 2. 3x - 8y ≤ 12 ; x = 0 3. - 10x - 2y > 7 ; x = -2 4. x + 5y ≥ 20 ; y = -1 5. 3x +2y < 21 ; y = 4 APPLICATION

22. • The solution of a linear equation is the set of points which lies on the line. • A solution of a linear inequality in two variables is an ordered pair (x, y) which makes the equation or inequality true. GENERALIZATION

23. EVALUATION Which of the following points is a solution to the following equations/ inequalities? Encircle your answer/s. 1. 3x + y > - 6 (3, 6) (4, 11) (2, 7) 2. y > 5x + 2 (2, -11) (-3, -8) (-5, 6) 3. -3x + 6y ≥ 10 (0, -3) (3, -5) (4, 8) 4. y ≤ 2x - 5 (2, -5) (9, -3) (12, 3) 5. 2y < 5x + 3 (3, 5) (2, 7) (3, 0)

24. ADDITIONAL ACTIVITY 1. a. Which ordered pair satisfies the inequality 3/2 x - 1/4y ≤ 1 ? a. (0, -5) b. (3, -5) c. (0, 1) d. (6, 0) b. Graph x + y = 6 in a Cartesian plane. Identify 5 points which are solution of the inequality x + y > 6, then plot them on the same plane. Make a conjecture about it. 2. Study how to graph linear inequality in two variables. Write the step by step process on your notebook.

25. Rewriting Linear Equations in Two Variables Ax+ By = C in the form y= mx + b and vice versa

26. Slide Title Product A Feature 1 Feature 2 Feature 3 Product B Feature 1 Feature 2 Feature 3

27. The process of rewriting linear equations in two variables Ax + By = C in the form y = mx + b can be done by solving y in terms of x. While, the process of rewriting linear equations in two variables, y = mx + b in the form Ax + By = C can be done by applying the different properties of real numbers and equations of Ax and By on the left- side of the equation and equate it to the constant term C on the right side.

28. Illustrative Examples: 1. 3x - 2y=4 - 2y = -3x+4 Solve y in terms of x - ( - 2y = -3x+4)-1/2 Apply MPE y = 3x/2 – 2 Simplify 2. y= 5x + 6 -5x + y =6 Sum of Ax and By on the left- hand side of the equation and equate it to the constant term C on the right- hand side - ( -5x + y = 6 ) Apply MPE 5x + y = 6 Simplify

29. ACTIVITY STANDARD FORM SLOPE-INTERCEPT FORM 1.) x – 3y =6 2.) 2x + 2y = -10 3.) y = -2x + 6 4.) y = 3x + 5

30. ACTIVITY STANDARD FORM SLOPE-INTERCEPT FORM 1.) x – 3y =6 (– 3y =-x + 6) ÷ -3 y = 3x + 2 2.) 2x + 2y = -10 (2y = -2x – 10) ÷ 2 y = -x – 5 y = -2x + 6 2x + y = 6 3.) y = -2x + 6 y = 3x + 5 (-3x + y = 5) times -1 3x – y = -5 4.) y = 3x + 5

31. Illustrative Examples: 1. 3x - 2y=4 - 2y = -3x+4 Solve y in terms of x - ( - 2y = -3x+4)-1/2 Apply MPE y = x – 2 Simplify 2. y= 5x + 6 -5x + y =6 Sum of Ax and By on the left- hand side of the equation and equate it to the constant term C on the right- hand side - ( -5x + y = 6 ) Apply MPE 5x + y = 6 Simplify

32. Two-Point Form   Where ( ) and ( ) are two points on the line  

33. Illustrative Examples: 1. 3x - 2y=4 - 2y = -3x+4 Solve y in terms of x - ( - 2y = -3x+4)-1/2 Apply MPE y = x – 2 Simplify 2. y= 5x + 6 -5x + y =6 Sum of Ax and By on the left- hand side of the equation and equate it to the constant term C on the right- hand side - ( -5x + y = 6 ) Apply MPE 5x + y = 6 Simplify

34. Illustrative Examples: 1. 3x - 2y=4 - 2y = -3x+4 Solve y in terms of x - ( - 2y = -3x+4)-1/2 Apply MPE y = x – 2 Simplify 2. y= 5x + 6 -5x + y =6 Sum of Ax and By on the left- hand side of the equation and equate it to the constant term C on the right- hand side - ( -5x + y = 6 ) Apply MPE 5x + y = 6 Simplify Point (4, 5) Point (-7, -17)