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Simultaneous Equations

Simultaneous Equations

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Simultaneous Equations

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  1. S5 Int2 Simultaneous Equations Solving Sim. Equations Graphically Graphs as Mathematical Models Solving Simple Sim. Equations by Substitution www.mathsrevision.com Solving Simple Sim. Equations by elimination Solving harder type Sim. equations Created by Mr. Lafferty Maths Department

  2. S5 Int2 Starter Questions www.mathsrevision.com Created by Mr. Lafferty Maths Department

  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. • Find the coordinates where • 2 lines intersect ( meet) www.mathsrevision.com Created by Mr. Lafferty Maths Department

  4. Q. Find the equation of each line. (1,3) Q. Write down the coordinates where they meet. Created by Mr. Lafferty Maths Department

  5. Q. Find the equation of each line. Q. Write down the coordinates where they meet. (-0.5,-0.5) Created by Mr. Lafferty Maths Department

  6. Q. Plot the lines. (1,1) Q. Write down the coordinates where they meet. Created by Mr. Lafferty Maths Department

  7. Now try Exercise 2 Ch7 (page 84 ) www.mathsrevision.com Created by Mr. Lafferty Maths Department

  8. S5 Int2 Starter Questions 8cm www.mathsrevision.com 5cm Created by Mr. Lafferty Maths Department

  9. Simultaneous Equations S5 Int2 Straight Lines Learning Intention Success Criteria • To use graphical methods to solve real-life mathematical models • Draw line graphs given a table of points. • Find the coordinates where • 2 lines intersect ( meet) www.mathsrevision.com Created by Mr. Lafferty Maths Department

  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 Created by Mr. Lafferty Maths Department

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

  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. Created by Mr. Lafferty Maths Department

  13. Now try Exercise 3 Ch7 (page 85 ) www.mathsrevision.com Created by Mr. Lafferty Maths Department

  14. S5 Int2 Starter Questions www.mathsrevision.com Created by Mr. Lafferty Maths Department

  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. www.mathsrevision.com Created by Mr. Lafferty Maths Department

  16. Example 1 Solve the equations y = 2x y = x+1 by substitution Created by Mr. Lafferty Maths Department

  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) Created by Mr. Lafferty Maths Department

  18. Example 1 Solve the equations y = x + 1 x + y = 4 by substitution (1.5, 2.5) Created by Mr. Lafferty Maths Department

  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 Created by Mr. Lafferty Maths Department

  20. Now try Ex 4 Ch7 (page88 ) www.mathsrevision.com Created by Mr. Lafferty Maths Department

  21. S5 Int2 Starter Questions www.mathsrevision.com Created by Mr. Lafferty Maths Department

  22. Simultaneous Equations S5 Int2 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 www.mathsrevision.com Created by Mr. Lafferty Maths Department

  23. Example 1 Solve the equations x + 2y = 14 x + y = 9 by elimination Created by Mr. Lafferty Maths Department

  24. 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 Created by Mr. Lafferty Maths Department

  25. Step 3: Sub into one of the equations to get other variable Substitute y = 5 in (B) x + y = 9 (B) 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) Created by Mr. Lafferty Maths Department

  26. Example 2 Solve the equations 2x - y = 11 x - y = 4 by elimination Created by Mr. Lafferty Maths Department

  27. 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 Created by Mr. Lafferty Maths Department

  28. Step 3: Sub into one of the equations to get other variable Substitute x = 7 in (B) x - y = 4 (B) 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) Created by Mr. Lafferty Maths Department

  29. Example 3 Solve the equations 2x - y = 6 x + y = 9 by elimination Created by Mr. Lafferty Maths Department

  30. 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) x = 15 ÷ 3 = 5 3x = 15 Created by Mr. Lafferty Maths Department

  31. Step 3: Sub into one of the equations to get other variable Substitute x = 5 in (B) x + y = 9 (B) 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) Created by Mr. Lafferty Maths Department

  32. Now try Ex 5A Ch7 (page89 ) www.mathsrevision.com Created by Mr. Lafferty Maths Department

  33. S5 Int2 Starter Questions www.mathsrevision.com Created by Mr. Lafferty Maths Department

  34. Simultaneous Equations S5 Int2 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. www.mathsrevision.com Created by Mr. Lafferty Maths Department

  35. Example 1 Solve the equations 2x + y = 9 x - 3y = 1 by elimination Created by Mr. Lafferty Maths Department

  36. Step 1: Label the equations 2x + y = 9 (A) x -3y = 1 (B) Step 2: Decide what you want to eliminate Adding Eliminate y by : (A) x3 2x + y = 9 x -3y = 1 6x + 3y = 27 (C) x - 3y = 1(D) (B) x1 7x = 28 x = 28 ÷ 7 = 4 Created by Mr. Lafferty Maths Department

  37. 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 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) Created by Mr. Lafferty Maths Department

  38. Example 2 Solve the equations 3x + 2y = 13 2x + y = 8 by elimination Created by Mr. Lafferty Maths Department

  39. Step 1: Label the equations 3x + 2y = 13 (A) 2x + y = 8 (B) Step 2: Decide what you want to eliminate Subtract Eliminate y by : (A) x1 3x + 2y = 13 2x + y = 8 3x + 2y = 13 (C) 4x + 2y = 16(D) (B) x2 -x = -3 x = 3 Created by Mr. Lafferty Maths Department

  40. 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 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) Created by Mr. Lafferty Maths Department

  41. Now try Ex 5B Ch7 (page90 ) www.mathsrevision.com Created by Mr. Lafferty Maths Department