1. linear equations in two variables introduction, solution and graphing

linear equations in two variables

1. What are linear equations in two variables? Learning Objective

2. To interpret basic concepts To find solution of equation with 2 variables To graph linear equation with 2 variables Learning Outcomes

3. Linear Equations in Two Variables Let’s look at a few equations. 7 x + 5 = 33 x + 5 y = 8 What do you notice?

4. Linear Equations in Two Variables Let’s look at a few equations. 7 x + 5 = 33 x + 5 y = 8 Both are linear equations!

5. Linear Equations in Two Variables Let’s look at a few equations. 7 x + 5 = 33 only one variable x x + 5 y = 8 two variables x and y Both are linear equations!

6. Linear Equations in Two Variables Let’s look at a few equations. x + 5 y = 8 two variables x and y This is a linear equation in two variables !

7. Linear Equations in Two Variables Let’s look at a few equations. x + 5 y = 8 two variables x and y Every linear equation in two variables has a standard form . This is a linear equation in two variables !

8. Linear Equations in Two Variables Let’s look at a few equations. x + 5 y = 8 two variables x and y Every linear equation in two variables has a standard form .  a x + b y + c = 0 This is a linear equation in two variables !

9. Linear Equations in Two Variables Let’s look at a few equations. x + 5 y = 8 two variables x and y Every linear equation in two variables has a standard form .  a x + b y + c = 0; Where x and y are variables . This is a linear equation in two variables !

10. Linear Equations in Two Variables Let’s look at a few equations. x + 5 y = 8 two variables x and y Every linear equation in two variables has a standard form .  a x + b y + c = 0; Where x and y are variables, And a , b and c are constants . Can you arrange the given equation in standard form ?

11. Linear Equations in Two Variables Let’s look at a few equations. x + 5 y = 8 two variables x and y x + 5 y – 8 = 0 Every linear equation in two variables has a standard form .  a x + b y + c = 0; Where x and y are variables, And a , b and c are constants . That’s the standard form .

12. Linear Equations in Two Variables Can you write this equation in standard form? 2 x = 3 Hint : The RHS should be 0.

13. Linear Equations in Two Variables How will we write this equation in two variables? 2 x = 3 2 x - 3 = 0 Hint : Add a variable such that it’s value is 0.

14. Linear Equations in Two Variables How will we write this equation in two variables? 2 x = 3 2 x + 0 y - 3 = 0 Why cannot the variable be x ?

15. Linear Equations in Two Variables How will we write this equation in two variables? 2 x = 3 2 x + 0 y - 3 = 0 That’s our standard form !

16. To interpret basic concepts To find solution of equation with 2 variables To graph linear equation with 2 variables Learning Outcomes How confident do you feel?

17. To interpret basic concepts To find solution of equation with 2 variables To graph linear equation with 2 variables Learning Outcomes How confident do you feel?

18. Linear Equations in Two Variables Let’s consider this equation! 2 x + 3 y = 12

19. Linear Equations in Two Variables Let’s consider this equation! 2 x + 3 y = 12 Let’s say value of x is 2. What will the value of y be?

20. Linear Equations in Two Variables Let’s consider this equation! 2 x + 3 y = 12 Let’s say value of x is 2. ∴ 2( 2 ) + 3 y = 12 Let’s substitute the value of x .

21. Linear Equations in Two Variables Let’s consider this equation! 2 x + 3 y = 12 Let’s say value of x is 2. ∴ 2( 2 ) + 3 y = 12 ∴ 4 + 3 y = 12 Let’s move 4 to the RHS!

22. Linear Equations in Two Variables Let’s consider this equation! 2 x + 3 y = 12 Let’s say value of x is 2. ∴ 2( 2 ) + 3 y = 12 ∴ 4 + 3 y = 12 ∴ 3 y = 12 - 4 Let’s move 4 to the RHS!

23. Linear Equations in Two Variables Let’s consider this equation! 2 x + 3 y = 12 Let’s say value of x is 2. ∴ 2( 2 ) + 3 y = 12 ∴ 4 + 3 y = 12 ∴ 3 y = 8 What is the value of y ?

24. Linear Equations in Two Variables Let’s consider this equation! 2 x + 3 y = 12 Let’s say value of x is 2. ∴ 2( 2 ) + 3 y = 12 ∴ 4 + 3 y = 12 ∴ 3 y = 8 ∴ y =   What is the value of y ?

25. Linear Equations in Two Variables Let’s consider this equation! 2 x + 3 y = 12 Let’s say value of x is 2. ∴ 2( 2 ) + 3 y = 12 ∴ 4 + 3 y = 12 ∴ 3 y = 8 ∴ y = ∴ x = 2 & y =   This is the solution of this equation!

26. Linear Equations in Two Variables Let’s consider this equation! 2 x + 3 y = 12 Let’s say value of x is 2. ∴ 2( 2 ) + 3 y = 12 ∴ 4 + 3 y = 12 ∴ 3 y = 8 ∴ y = ∴ x = 2 & y =   The solution is also written as ( 2 , ) .  

27. Linear Equations in Two Variables Let’s consider this equation! 2 x + 3 y = 12 Solutions : ( 2 , )   Can you write a few more solutions ?

28. Linear Equations in Two Variables Let’s consider this equation! 2 x + 3 y = 12 Solutions : ( 2 , ) ( , 4 ) ( 6 , ) ( 1 , )….   What do you notice?

29. Linear Equations in Two Variables Let’s consider this equation! 2 x + 3 y = 12 Solutions : ( 2 , ) ( , 4 ) ( 6 , ) ( 1 , )….   There are too many solutions .

30. Linear Equations in Two Variables Let’s consider this equation! A linear equation in two variables has _____________ solutions .

31. Linear Equations in Two Variables Let’s consider this equation! A linear equation in two variables has infinitely many solutions .

32. Linear Equations in Two Variables Find four different solutions for this equation. x + 2 y = 6 Hint : Always use 0 as the value of the variables.

33. Linear Equations in Two Variables Find four different solutions for this equation. x + 2 y = 6 Solutions : ( 2 , 2 ) ( , 3 ) ( 6 , ) ( 4 , 1 )…. Hint : Always use 0 as the value of the variables.

34. To interpret basic concepts To find solution of equation with 2 variables To graph linear equation with 2 variables Learning Outcomes How confident do you feel?

35. To interpret basic concepts To find solution of equation with 2 variables To graph linear equation with 2 variables Learning Outcomes How confident do you feel?

36. Linear Equations in Two Variables Let’s draw a graph for this equation! x + 2 y = 6 Solutions : ( , 3 ) ( 2 , 2 ) ( 4 , 1 ) ( 6 , )…. We will use each solution as a single point!

37. Linear Equations in Two Variables Let’s draw a graph for this equation! x + 2 y = 6 Now, let’s draw a graph! x y Point 3 (0,3) 2 2 (2,2) 4 1 (4,1) 6 (6,0)

38. Linear Equations in Two Variables Let’s draw a graph for this equation! x + 2 y = 6 x y Point 3 (0,3) 2 2 (2,2) 4 1 (4,1) 6 (6,0) Y X O Let’s mark the co-ordinates !

39. Linear Equations in Two Variables Let’s draw a graph for this equation! x + 2 y = 6 x y Point 3 (0,3) 2 2 (2,2) 4 1 (4,1) 6 (6,0) Y X O Let’s write the co-ordinates !

40. Linear Equations in Two Variables Let’s draw a graph for this equation! x + 2 y = 6 x y Point 3 (0,3) 2 2 (2,2) 4 1 (4,1) 6 (6,0) Y X O Let’s mark the points ! 1 2 3 4 5 6 7 1 2 3 4 5 6 7 8 9

41. Linear Equations in Two Variables Let’s draw a graph for this equation! x + 2 y = 6 x y Point 3 A (0,3) 2 2 (2,2) 4 1 (4,1) 6 (6,0) Y X O Let’s mark the points ! 1 2 3 4 5 6 7 1 2 3 4 5 6 7 8 9 A

42. Linear Equations in Two Variables Let’s draw a graph for this equation! x + 2 y = 6 x y Point 3 A (0,3) 2 2 B (2,2) 4 1 (4,1) 6 (6,0) Y X O Let’s mark the points ! 1 2 3 4 5 6 7 1 2 3 4 5 6 7 8 9 B A

43. Linear Equations in Two Variables Let’s draw a graph for this equation! x + 2 y = 6 x y Point 3 A (0,3) 2 2 B (2,2) 4 1 C (4,1) 6 (6,0) Y X O Let’s mark the points ! 1 2 3 4 5 6 7 1 2 3 4 5 6 7 8 9 C B A

44. Linear Equations in Two Variables Let’s draw a graph for this equation! x + 2 y = 6 x y Point 3 A (0,3) 2 2 B (2,2) 4 1 C (4,1) 6 D (6,0) Y X O Let’s mark the points ! 1 2 3 4 5 6 7 1 2 3 4 5 6 7 8 9 D C B A

45. Linear Equations in Two Variables Let’s draw a graph for this equation! x + 2 y = 6 x y Point 3 A (0,3) 2 2 B (2,2) 4 1 C (4,1) 6 D (6,0) Y X O This is the graph for x + 2 y = 6. 1 2 3 4 5 6 7 1 2 3 4 5 6 7 8 9 A B C D

46. To interpret basic concepts To find solution of equation with 2 variables To graph linear equation with 2 variables Learning Outcomes How confident do you feel?

47. To interpret basic concepts To find solution of equation with 2 variables To graph linear equation with 2 variables Learning Outcomes How confident do you feel?

48. What are linear equations in two variables? Learning Objective

49. Linear Equations in Two Variables Learning Activity Draw the graph of x + y = 7.