1. 26-Dec-15 Solving Sim. Equations Graphically Solving Simple Sim. Equations by Substitution Simultaneous Equations www.mathsrevision.com Solving Simple Sim. Equations by elimination Solving harder type Sim. equations S5 Int2 Graphs as Mathematical Models

2. 26-Dec-15 Starter Questions Starter Questions www.mathsrevision.com S5 Int2

3. 26-Dec-15 www.mathsrevision.com Learning Intention Success Criteria 1.To solve simultaneous equations using graphical methods. Simultaneous Equations 1.Interpret information from a line graph. 2.Plot line equations on a graph. 3.Find the coordinates were 2 lines intersect ( meet) Straight Lines S5 Int2

4. www.mathsrevision.com Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 Q. Find the equation of each line. (1,3) Q. Write down the coordinates were they meet.

5. www.mathsrevision.com Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 Q. Find the equation of each line. (-0.5,-0.5) Q. Write down the coordinates where they meet.

6. www.mathsrevision.com Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 Q. Plot the lines. (1,1) Q. Write down the coordinates where they meet.

7. www.mathsrevision.com Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 Now try Exercise 2 Ch7 (page 84 ) www.mathsrevision.com

8. 26-Dec-15 Starter Questions Starter Questions www.mathsrevision.com S5 Int2 5cm 8cm

9. 26-Dec-15 www.mathsrevision.com Learning Intention Success Criteria 1.To use graphical methods to solve real-life mathematical models Simultaneous Equations 1.Draw line graphs given a table of points. 2.Find the coordinates were 2 lines intersect ( meet) Straight Lines S5 Int2

10. www.mathsrevision.com Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 We can use straight line theory to work out real-life problems especially useful when trying to work out hire charges. Q.I need to hire a car for a number of days. Below are the hire charges charges for two companies. Complete tables and plot values on the same graph. 160180200 180240300

11. www.mathsrevision.com Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 Days Total Cost £ A r n o l d S w i n t o n Summarise data ! Who should I hire the car from? Up to 2 days Swinton Over 2 days Arnold

12. www.mathsrevision.com Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 Key steps 1. Fill in tables 2. Plot points on the same graph ( pick scale carefully) 3. Identify intersection point ( where 2 lines meet) 4. Interpret graph information.

13. www.mathsrevision.com Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 Now try Exercise 3 Ch7 (page 85 ) www.mathsrevision.com

14. 26-Dec-15 Starter Questions Starter Questions www.mathsrevision.com S5 Int2

15. 26-Dec-15 www.mathsrevision.com Learning Intention Success Criteria 1.To solve pairs of equations by substitution. Simultaneous Equations 1.Apply the process of substitution to solve simple simultaneous equations. Straight Lines S5 Int2

16. www.mathsrevision.com Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 Example 1 Solve the equations y = 2x y = x+1 by substitution

17. www.mathsrevision.com Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 At the point of intersection y coordinates are equal: 2x = x+1 Rearranging we get : 2x - x = 1 x = 1 Finally : Sub into one of the equations to get y value y = 2x = 2 x 1 = 2 OR y = x+1 = 1 + 1 = 2 so we have y = 2x y = x+1 The solution is x = 1 y = 2 or (1,2)

18. www.mathsrevision.com Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 Example 1 Solve the equations y = x + 1 x + y = 4 by substitution (1.5, 2.5)

19. www.mathsrevision.com Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 At the point of intersection y coordinates are equal: x+1 = -x+4 Rearranging we get : 2x = 4 - 1 2x = 3 Finally : Sub into one of the equations to get y value y = x +1 = 1.5 + 1 = 2.5 y = -x+4 = -1.5 + 4 = 2.5 so we have y = x +1 y =-x+ 4 The solution is x = 1.5 y = 2.5 (1.5,2.5) x = 3 ÷ 2 = 1.5 OR

20. www.mathsrevision.com Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 Now try Ex 4 Ch7 (page88 ) www.mathsrevision.com

21. 26-Dec-15 Starter Questions Starter Questions www.mathsrevision.com S5 Int2

22. 26-Dec-15 www.mathsrevision.com Learning Intention Success Criteria 1.To solve simultaneous equations of 2 variables. Simultaneous Equations 1.Understand the term simultaneous equation. 2.Understand the process for solving simultaneous equation of two variables. 3.Solve simple equations Straight Lines S5 Int2

23. www.mathsrevision.com Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 Example 1 Solve the equations x + 2y = 14 x + y = 9 by elimination

24. www.mathsrevision.com Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 Step 1: Label the equations x + 2y = 14 (A) x + y = 9 (B) Step 2: Decide what you want to eliminate Eliminate x by subtracting (B) from (A) x + 2y = 14 (A) x + y = 9 (B) y = 5

25. www.mathsrevision.com Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 Step 3: Sub into one of the equations to get other variable Substitute y = 5 in (B) x + y = 9 (B) x + 5 = 9 The solution is x = 4 y = 5 Step 4:Check answers by substituting into both equations x = 9 - 5 x = 4 x + 2y = 14 x + y = 9 ( 4 + 10 = 14) ( 4 + 5 = 9)

26. www.mathsrevision.com Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 Example 2 Solve the equations 2x - y = 11 x - y = 4 by elimination

27. www.mathsrevision.com Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 Step 1: Label the equations 2x - y = 11 (A) x - y = 4 (B) Step 2: Decide what you want to eliminate Eliminate y by subtracting (B) from (A) 2x - y = 11 (A) x - y = 4 (B) x = 7

28. www.mathsrevision.com Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 Step 3: Sub into one of the equations to get other variable Substitute x = 7 in (B) x - y = 4 (B) 7 - y = 4 The solution is x =7 y =3 Step 4:Check answers by substituting into both equations y = 7 - 4 y = 3 2x - y = 11 x - y = 4 ( 14 - 3 = 11) ( 7 - 3 = 4)

29. www.mathsrevision.com Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 Example 3 Solve the equations 2x - y = 6 x + y = 9 by elimination

30. www.mathsrevision.com Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 Step 1: Label the equations 2x - y = 6 (A) x + y = 9 (B) Step 2: Decide what you want to eliminate Eliminate y by adding (A) from (B) 2x - y = 6 (A) x + y = 9 (B) 3x = 15 x = 15 ÷ 3 = 5

31. www.mathsrevision.com Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 Step 3: Sub into one of the equations to get other variable Substitute x = 5 in (B) x + y = 9 (B) 5 + y = 9 The solution is x = 5 y = 4 Step 4: Check answers by substituting into both equations y = 9 - 5 y = 4 2x - y = 6 x + y = 9 ( 10 - 4 = 6) ( 5 + 4 = 9)

32. www.mathsrevision.com Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 Now try Ex 5A Ch7 (page89 ) www.mathsrevision.com

33. 26-Dec-15 Starter Questions Starter Questions www.mathsrevision.com S5 Int2

34. 26-Dec-15 www.mathsrevision.com Learning Intention Success Criteria 1.To solve harder simultaneous equations of 2 variables. Simultaneous Equations 1.Apply the process for solving simultaneous equations to harder examples. Straight Lines S5 Int2

35. www.mathsrevision.com Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 Example 1 Solve the equations 2x + y = 9 x - 3y = 1 by elimination

36. www.mathsrevision.com Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 2x + y = 9 x -3y = 1 Step 1: Label the equations 2x + y = 9(A) x -3y = 1(B) Step 2: Decide what you want to eliminate Eliminate y by : 7x = 28 6x + 3y = 27 (C) x - 3y = 1 (D) x = 28 ÷ 7 = 4 Adding (A) x3 (B) x1

37. www.mathsrevision.com Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 Step 3: Sub into one of the equations to get other variable Substitute x = 4 in equation (A) 2 x 4 + y = 9 y = 9 – 8 The solution is x = 4 y = 1 Step 4: Check answers by substituting into both equations y = 1 2x + y = 9 x -3y = 1 ( 8 + 1 = 9) ( 4 - 3 = 1)

38. www.mathsrevision.com Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 Example 2 Solve the equations 3x + 2y = 13 2x + y = 8 by elimination

39. www.mathsrevision.com Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 3x + 2y = 13 2x + y = 8 Step 1: Label the equations 3x + 2y = 13(A) 2x + y = 8(B) Step 2: Decide what you want to eliminate Eliminate y by : -x = -3 3x + 2y = 13 (C) 4x + 2y = 16 (D) x = 3 Subtract (A) x1 (B) x2

40. www.mathsrevision.com Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 Step 3: Sub into one of the equations to get other variable Substitute x = 3 in equation (B) 2 x 3 + y = 8 y = 8 – 6 The solution is x = 3 y = 2 Step 4: Check answers by substituting into both equations y = 2 3x + 2y = 13 2x + y = 8 ( 9 + 4 = 13) ( 6 + 2 = 8)

41. www.mathsrevision.com Simultaneous Equations S5 Int2 Straight Lines 26-Dec-15 Now try Ex 5B Ch7 (page90 ) www.mathsrevision.com