1. Chapter 17 Simultaneous Equations

2. Learning Objectives Use graphs to solve simultaneous equations Show that certain simultaneous equations have no solution Solve simultaneous equation by elimination Solve simultaneous equations by substitution Solve practical problems with simultaneous equations

3. Using Graphs to Solve Simultaneous Equations Pick 3 x-points for each of the equations Work out the y-values Draw each of the equations Examples 1. Solve x + 2y = 5 x – 2y = 1

4. Using Graphs to Solve Simultaneous Equations 2. Solve x + 2y = 5 x – 2y = 1

5. Using Graphs to Show Simultaneous Equations Can Have NO Solution If the two graphs do not cross each other there is no solution Examples 1. Show that y – 2x = 4 2y = 4x – 1 have no solution

6. Using Algebra to Show that Simultaneous Equations Can Have NO Solution If the two graphs have the same gradient then they are parallel and do not cross so they do not have a solution Re-arrange the equation to put it in the form y = mx + c (remember that m = grad)

7. Examples 1. Show that y – 2x = 4 and 2y = 4x – 1 do not have any solution. 2. Show that y + 1 = -5x and 2y + 10x = 5 do not have any solution

8. The Elimination Method Add or subtract the equation to eliminate either the x or the y Examples 1. Solve 2x + 3y = 9 and 2x + y = 7 2. Solve x + 2y = 5 and x – 2y = 1 3. Solve 2x – y = 1 and 3x + y = 9

9. Harder Elimination If neither the x’s nor the y’s are the same make them the same by multiplying either 1 or 2 of the equations Examples 1. 5x + 2y = 11 and 3x – 4y = 4 2. 3x + 7y = -2 and 4x + 9 = -3y 3. 9x = 4y – 20 and 5x = 6y - 13

10. The Substitution Method Re-arrange so that one of the equations has either x= or y= Then substitute this into the other equation Examples 1. Solve 5x + y = 9 and y = 4x 2. Solve x + 4y = 32 and x = 2y - 4

11. Practical Problems Examples 1. Billy buys 5 1 st class stamps and 3 2 nd class stamps for £1.93. Jane buys 3 1 st class and 5 2 nd class stamps for £1.83. How much is a 1 st class and 2 nd class stamp?

12. Practical Problems 2. Micro-scooters costs £x each and pogo sticks cost £y each. 2 micro-scooters and 4 pogo sticks cost £65 1 micro-scooter and 3 pogo sticks cost £40 Find the value of x and y