Chapter 17 Simultaneous Equations Learning Objectives l l

Chapter 17 Simultaneous Equations

Learning Objectives l l l 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

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

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

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

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

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

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

Harder Elimination l 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. 5 x + 2 y = 11 and 3 x – 4 y = 4 2. 3 x + 7 y = -2 and 4 x + 9 = -3 y 3. 9 x = 4 y – 20 and 5 x = 6 y - 13

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

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?

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