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# Simultaneous Equations

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1. Simultaneous Equations Solving Sim. Equations Graphically Graphs as Mathematical Models Solving Simple Sim. Equations by Substitution Solving Simple Sim. Equations by elimination Solving harder type Sim. equations

2. Starter Questions

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

4. Q. Find the equation of each line. (1,3) Q. Write down the coordinates were they meet.

5. Q. Find the equation of each line. Q. Write down the coordinates where they meet. (-0.5,-0.5)

6. Q. Plot the lines. (1,1) Q. Write down the coordinates where they meet.

7. Now try Exercise 2 Ch7 (page 84 )

8. S5 Int2 Starter Questions 8cm 5cm

9. Simultaneous Equations Straight Lines Learning Intention Success Criteria • To use graphical methods to solve real-life mathematical models • Draw line graphs given a table of points. • 2. Find the coordinates were 2 lines intersect ( meet)

10. 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. 160 180 200 180 240 300

11. Summarise data ! Who should I hire the car from? Arnold Total Cost £ Up to 2 days Swinton Over 2 days Arnold Swinton Days

12. 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. Now try Exercise 3 Ch7 (page 85 )

14. S5 Int2 Starter Questions

15. Simultaneous Equations S5 Int2 Straight Lines Learning Intention Success Criteria • To solve pairs of equations by substitution. 1. Apply the process of substitution to solve simple simultaneous equations.

16. Example 1 Solve the equations y = 2x y = x+1 by substitution

17. y = 2x y = x+1 At the point of intersection y coordinates are equal: 2x = x+1 so we have 2x - x = 1 Rearranging we get : x = 1 Finally : Sub into one of the equations to get y value y = 2x = 2 x 1 = 2 y = x+1 = 1 + 1 = 2 OR The solution is x = 1 y = 2 or (1,2)

18. Example 1 Solve the equations y = x + 1 x + y = 4 by substitution (1.5, 2.5)

19. y = x +1 y =-x+ 4 The solution is x = 1.5 y = 2.5 (1.5,2.5) At the point of intersection y coordinates are equal: x+1 = -x+4 so we have 2x = 4 - 1 Rearranging we get : 2x = 3 x = 3 ÷ 2 = 1.5 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 OR

20. Now try Ex 4 Ch7 (page88 )

21. Starter Questions

22. Simultaneous Equations Straight Lines Learning Intention Success Criteria • To solve simultaneous equations of 2 variables. • Understand the term simultaneous equation. • Understand the process for solving simultaneous equation of two variables. • 3. Solve simple equations

23. Example 1 Solve the equations x + 2y = 14 x + y = 9 by elimination

24. Step 1: Label the equations x + 2y = 14 (1) x + y = 9 (2) Step 2: Decide what you want to eliminate Eliminate x by subtracting (2) from (1) x + 2y = 14 (1) x + y = 9 (2) y = 5

25. Step 3: Sub into one of the equations to get other variable Substitute y = 5 in (2) x + y = 9 (2) x + 5 = 9 x = 9 - 5 The solution is x = 4 y = 5 x = 4 Step 4: Check answers by substituting into both equations ( 4 + 10 = 14) x + 2y = 14 x + y = 9 ( 4 + 5 = 9)

26. Example 2 Solve the equations 2x - y = 11 x - y = 4 by elimination

27. Step 1: Label the equations 2x - y = 11 (1) x - y = 4 (2) Step 2: Decide what you want to eliminate Eliminate y by subtracting (2) from (1) 2x - y = 11 (1) x - y = 4 (2) x = 7

28. Step 3: Sub into one of the equations to get other variable Substitute x = 7 in (2) x - y = 4 (2) 7 - y = 4 y = 7 - 4 The solution is x =7 y =3 y = 3 Step 4: Check answers by substituting into both equations ( 14 - 3 = 11) 2x - y = 11 x - y = 4 ( 7 - 3 = 4)

29. Example 3 Solve the equations 2x - y = 6 x + y = 9 by elimination

30. Step 1: Label the equations 2x - y = 6 (1) x + y = 9 (2) Step 2: Decide what you want to eliminate Eliminate y by adding (1) and (2) 2x - y = 6 (1) x + y = 9 (2) x = 15 ÷ 3 = 5 3x = 15

31. Step 3: Sub into one of the equations to get other variable Substitute x = 5 in (2) x + y = 9 (2) 5 + y = 9 y = 9 - 5 The solution is x = 5 y = 4 y = 4 Step 4: Check answers by substituting into both equations ( 10 - 4 = 6) 2x - y = 6 x + y = 9 ( 5 + 4 = 9)

32. Now try Ex 5A Ch7 (page89 )

33. Starter Questions

34. Simultaneous Equations Straight Lines Learning Intention Success Criteria • To solve harder simultaneous equations of 2 variables. 1. Apply the process for solving simultaneous equations to harder examples.

35. Example 1 Solve the equations 2x + y = 9 x - 3y = 1 by elimination

36. Step 1: Label the equations 2x + y = 9 (1) x -3y = 1 (2) Step 2: Decide what you want to eliminate Adding Eliminate y by : (1) x3 2x + y = 9 x -3y = 1 6x + 3y = 27 (3) x - 3y = 1(4) (2) x1 7x = 28 x = 28 ÷ 7 = 4

37. Step 3: Sub into one of the equations to get other variable Substitute x = 4 in equation (1) 2 x 4 + y = 9 y = 9 – 8 y = 1 The solution is x = 4 y = 1 Step 4: Check answers by substituting into both equations ( 8 + 1 = 9) 2x + y = 9 x -3y = 1 ( 4 - 3 = 1)

38. Example 2 Solve the equations 3x + 2y = 13 2x + y = 8 by elimination

39. Step 1: Label the equations 3x + 2y = 13 (1) 2x + y = 8 (2) Step 2: Decide what you want to eliminate Subtract Eliminate y by : (1) x1 3x + 2y = 13 2x + y = 8 3x + 2y = 13 (3) 4x + 2y = 16(4) (2) x2 -x = -3 x = 3

40. Step 3: Sub into one of the equations to get other variable Substitute x = 3 in equation (2) 2 x 3 + y = 8 y = 8 – 6 y = 2 The solution is x = 3 y = 2 Step 4: Check answers by substituting into both equations ( 9 + 4 = 13) 3x + 2y = 13 2x + y = 8 ( 6 + 2 = 8)

41. Now try Ex 5B Ch7 (page90 )