Simultaneous Equations. Powerpoint presentation
- 1. Starter Solve: a) 4x = -16 b) x + 5 = -6 c) 2x - 3 = 11 d) 8 – 6x = 26 Substitute for x = -1, y = 5 e) 3x + 2y f) 4y – 6x x = -4 x = -11 x = 7 x = -3 -3 + 10 = 7 20 + 6 = 26
- 2. How much do the burgers cost? How much do the chips cost? + + = £12 + = £9
- 3. How much do the burgers cost? How much do the chips cost? + + + = £12 + = £8
- 4. How much do the burgers cost? How much do the chips cost? + = £8 + = £5 +
- 5. Simultaneous Equations Scale up each term in one or both equations to make the coefficients the same for either the x terms or the y terms. Subtract if the signs in front of these are the same. Add if the signs in front of these are the different.
- 6. 5x + y = 20 …(1) 2x + y = 11 …(2) Scale up (if necessary) Add or subtract (to eliminate) Solve (to find x) Substitute in (to find y) 3x - = 9 x = 3 5x + y = 20 15 + y = 20 y = 5
- 7. 7x + 2y = 32 …(1) 3x – 2y = 8 …(2) Scale up (if necessary) Solve (to find x) Substitute in (to find y) 10x + = 40 x = 4 7x + 2y = 32 28 + 2y = 32 2y = 4 Add or subtract (to eliminate) y = 2
- 8. How much do the burgers cost? How much do the chips cost? + = £20 + = £8.50 + + +
- 9. 12x – 2y = 8 …(1) 5x + y = 18 …(2) Scale up (if necessary) Solve (to find x) Substitute in (to find y) 22x + = 44 x = 2 12x – 2y = 8 24 – 2y = 8 -2y = -16 Add or subtract (to eliminate) y = 8 12x – 2y = 8 10x + 2y = 36 x1 x2
- 10. 7x – 3y = 29 …(1) 2x + 5y = 20 …(2) Scale up (if necessary) Solve (to find y) Substitute in (to find x) -41y - = -82 y = 2 7x – 3y = 29 7x – 6 = 29 7x = 35 Add or subtract (to eliminate) x = 5 14x – 6y = 58 14x + 35y = 140 x2 x7
- 11. How much do the burgers cost? How much do the chips cost? + = £20 + = £8.50 + + +
- 12. Answers 1) x = 2, y = 3 2) x = 2, y = 4 3) x = 4, y = -1 4) x = 6, y = -5 5) x = 4.5, y = -3 6) x = -3, y = 5 7) x = 3, y = -0.5 8) x = -2, y = -5
- 13. Show me a pair of simultaneous equations where x = 3 and y = 2 Show me a pair of simultaneous equations where x = ½ and y = -4
- 14. A cinema sells adult tickets and child tickets. The total cost of 3 adult tickets and 1 child ticket is £30. The total cost of 1 adult ticket and 3 child tickets is £22. Work out the cost of an adult ticket and the cost of a child ticket. adult ticket £............................................... child ticket £............................................... (Total for Question is 4 marks)
- 15. Starter Equation Gradient y – intercept y = 3x + 2 2 7 y = 3x 3 y = -4x + 5 ½ 4 2y = 4x + 6 Copy and complete the following table:
- 16. Equation Gradient y – intercept y = 3x + 2 3 2 y = 2x + 7 2 7 y = 3x 3 0 y = -4x + 5 -4 5 y = ½x + 4 ½ 4 2y = 4x + 6 2 3 Answers
- 17. We are learning to solve simultaneous equations graphically.
- 18. y = mx + c m is the gradient, or the slope of the graph c is the y- intercept, or where the graph cuts the y-axis Remember:
- 19. Solve the simultaneous equations y = 2x + 1 and y = 3 graphically: Start by sketching y = 2x + 1 Start at 1 on the y-axis. For every 1 across, go up 2. Join with a straight line.
- 20. Solve the simultaneous equations y = 2x + 1 and y = 3 graphically: Start by sketching y = 2x + 1 Start at 1 on the y-axis. For every 1 across, go up 2. Join with a straight line.
- 21. Solve the simultaneous equations y = 2x + 1 and y = 3 graphically: The solution is the coordinate where the graphs cross. (1, 3) So x = 1 and y = 3.
- 22. Solve the simultaneous equations y = 3x + 2 and y = 6 – x graphically: Start by sketching y = 3x + 2 Start at 2 on the y-axis. For every 1 across, go up 3. Join with a straight line.
- 23. Solve the simultaneous equations y = 3x + 2 and y = 6 – x graphically: Now sketch y = 6 – x. Start at 6 on the y-axis. For every 1 across, go down 1. Join with a straight line.
- 24. Solve the simultaneous equations y = 3x + 2 and y = 6 – x graphically: The solution is the coordinate where the graphs cross. (1, 5) So x = 1 and y = 5.
- 25. Show me a pair of simultaneous equations with a solution at (5, 2). True/Never/Sometimes: All linear graphs intersect.
- 26. Answers x = 4, y = 8 x = -2, y = -6 x = 1, y = 3 x = 4, y = 0 x = 2, y = 1 x = 1, y = 4 x = 0, y = -2 x = 1, y = 1 x = 4, y = 9 x = 6, y = -6 x = 1, y = 1 x = -1, y = 1
- 27. Starter Solve the following, giving you answers to 2 d.p. where necessary: a) x² - x – 12 = 0 b) 6x² - x – 15 = 0 c) 3x² + 2x – 9 = 0 d) 4x² - 6x – 2 = 0 x = -3, 4 x = 1.67, -1.5 x = 1.43, -2.10 x = 1.78, -0.28
- 28. Simultaneous Equations (where one is linear and one is a quadratic) Linear graph Quadratic graph Solutions
- 29. Simultaneous Equations (where one is linear and one is a quadratic) y = 2x² + 6x + 4 y = -9x - 24 -9x - 24 = 2x² + 6x + 4 If they are both equal to y, they are equal to each other. Manipulate equation so it equals 0. 0 = 2x² + 15x + 28 Solve to find x. 0 = (2x + 7)(x + 4) x = -7/2 or -4 Substitute to find corresponding y values. y = -9x - 24 y = 63/2 - 24 y = 15/2 y = -9x - 24 y = 36 - 24 y = 12 Write as coordinates. (-7/2, 15/2) and (-4, 12)
- 30. Simultaneous Equations (where one is linear and one is a circle) Linear graph Circle Solutions
- 31. Simultaneous Equations (where one is linear and one is a circle) x² + y² = 16 y = 2x - 5 x² + (2x – 5)² = 16 Substitute the second equation for y in the first equation. Expand the brackets and manipulate equation so it equals 0. x² + 4x² - 20x + 25 – 16 = 0 Solve to find x. 5x² - 20x + 9 = 0 x = 3.48 or 0.52 Substitute to find corresponding y values. y = 2x - 5 y = 2(3.48) - 5 y = 1.96 y = 2x - 5 y = 2(0.52) - 5 y = -3.96 Write as coordinates. (3.48, 1.96) and (0.52, -3.96)
- 32. Answers (3.85, 9.85) (-5, -3) (-2.85, 3.15) (3, 5) (-1, 3) (4/3, 2/3) (0.2, 4.2) (1, 1)