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Year 9/GCSE: Simultaneous Equations

Year 9/GCSE: Simultaneous Equations. Dr J Frost (jfrost@tiffin.kingston.sch.uk) . Last modified: 27 th August 2013. How many solutions for x and y?. For x. For y. ?. 2. ?. ∞. x 2 = 4. ?. 1. ?. ∞. x = 3. x + y = 9. ?. ∞. ?. ∞. x + y = 9 x – y = 1. ?. ?. 1. 1. 8 6 4

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Year 9/GCSE: Simultaneous Equations

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  1. Year 9/GCSE: Simultaneous Equations Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 27th August 2013

  2. How many solutions for x and y? For x For y ? 2 ? ∞ x2 = 4 ? 1 ? ∞ x = 3 x + y = 9 ? ∞ ? ∞ x + y = 9 x – y = 1 ? ? 1 1

  3. 8 6 4 2 -2 -4 -6 By using graphical methods, solve the simultaneous equations: x + y = 7 2x – y = -1 Solution: x = 2, y = 5 But why does finding the intersection of the lines give the solution? The line for each equation represents all the points (x,y) for which the equation is satisfied. Therefore, at the intersection(s), this gives the points for which both equations are satisfied. x + y = 7 ? -10 -8 -6 -4 -2 2 4 6 8 10 2x – y = -1 Click to sketch

  4. 8 6 4 2 -2 -4 -6 3x - y = -4 2x + y = -1 Solution: x = -1, y = 1 -10 -8 -6 -4 -2 2 4 6 8 10 3x – y = -4 2x + y = -1 Click to sketch

  5. Exercises Copy the axis provided, and sketch the given lines on them. Hence solve the simultaneous equations. Q2 Q1 5 4 3 2 1 5 4 3 2 1 1 2 3 4 5 1 2 3 4 5

  6. Thinking graphically… For two simultaneous equations, when would we have… ? Lines are parallel but not the same. 0 solutions for and ? Infinitely many solutions for and ? ? Lines are the same. e.g.

  7. Three methods of solving simultaneous equations graphically by substitution by elimination

  8. Solving by Substitution We currently have two equations both involving two variables. 3x – 2y = 0 2x + y = 7 If we had just one equation involving one variable, then we could easily solve! ? To get rid of the y in the first equation, rearrange the second so it’s in terms of y. 2x + y = 7 y = 7 – 2x 3x – 2y = 0 3x – 2(7-2x) = 0 3x – 14 + 4x = 0 7x – 14 = 0 7x = 14 x = 2 Then y = 3

  9. Your go… Answer: Answer: x = 2, y = 1 ? ?

  10. Exercises GCSE Rayner (Old Edition) Page 105 - Exercise 27 Odd numbered questions.

  11. Harder GCSE Exam Question The graph shows two points (1,7) and (3,175) on a line with equation: y = kax Determine k and a. (3,175) (1,7) Answer: a = 5, k = 1.4 ?

  12. Olympiad Puzzle Mars, his wife Venus and grandson Pluto have a combined age of 192. The ages of Mars and Pluto together total 30 years more than Venus’ age. The ages of Venus and Pluto together total 4 years more than Mars’s age. Find their three ages. (Hint: You can form 3 equations with 3 unknowns. Substitute one equation into the two others to get 2 equations with 2 unknowns, then solve as normal) ? ? ? Mars = 94, Venus = 81, Pluto = 17

  13. Three methods of solving simultaneous equations graphically by elimination by substitution

  14. Solving by Elimination We can solve some simultaneous equations by adding or subtracting the equations. 2x + y = 6 3x – y = 9 • To eliminate a variable: • One –ve, one +ve ADD • Both –ve or +ve  SUBTRACT + 5x = 15 x = 3 Substituting that back into either equation: 9 – y = 9 So y = 0

  15. Solving by Elimination ? • Solve in 2 different ways: • Eliminating x. • Eliminating y. ?

  16. Solving by Elimination ? ? ?

  17. Problem Solving using Simultaneous Equations A The angle at A is 12° greater than the angle at C. Find x and y. x 3y C x B Answer: x = 64, y = 17 1 3 ?

  18. Exercises GCSE Rayner (Old Edition) Page 105 - Exercise 28 Odd numbered questions.

  19. Olympiad Puzzle Mars, his wife Venus and grandson Pluto have a combined age of 192. The ages of Mars and Pluto together total 30 years more than Venus’ age. The ages of Venus and Pluto together total 4 years more than Mars’s age. Find their three ages. Solve by elimination this time! Try adding/subtracting equations. ? ? ? Mars = 94, Venus = 81, Pluto = 17

  20. Olympiad Puzzle James, Alison and Vivek go into a shop to buy some sweets. James spends £1 on four Fudge Bars, a Sparkle and a Chomper. Alison spends 70p on three Chompers, two Fudge Bars and a Sparkle. Vivek spends 50p on two Sparkles and a Fudge Bar. What is the cost of a Sparkle? Sparkle = 15p ?

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