Simultaneous Equations 6-Apr-17 Objective Solve a pairs of simultaneous equations like’ 3x + 2y = 20 4x – 4y = 10 and 3x + 2y = 36 2x + 3y = 29 © Brain-Cells: E.Resources Ltd. All Rights Reserved 24/11/09 www.brain-cells.co.uk

Here are some of the answers For an equation like x + 3 = 5, there is only one unknown value and this is x = 2 But an equation like x + y = 5 with two unknown values, has lots of (infinite) answers Here are some of the answers x + y = 5 x = 4 y = 1 4 + 1 = 5 x = 3 y = 2 3 + 2 = 5 x = 2 y = 3 2 + 3 = 5 x = 1 y = 4 1 + 4 = 5 © Brain-Cells: E.Resources Ltd. All Rights Reserved 24/11/09 www.brain-cells.co.uk

Pairs of equations like this are called simultaneous equations But if we add a second equation like this: x + y = 5 x – y = 1 there is only one value of x and one value of y that will fit into two equations 3 + 2 = 5 3 – 2 = 1 x = 3 and y = 1 Pairs of equations like this are called simultaneous equations © Brain-Cells: E.Resources Ltd. All Rights Reserved 24/11/09 www.brain-cells.co.uk

Middle signs are different so 3x + 2y = 16 (1) 2x - 2y = 4 (2) Middle signs are different so (1) + (2) will eliminate y First make sure that the middle numbers are the same and both are 2y, so this is OK. Look at the signs in front of the middle numbers. Same SUBTRACT Different ADD. Finally, replace the x with 4 in one of the equations to find y It’s a good idea to label the equations (1) and (2) 5x = 20 x = 20 ÷ 5 x = 4 3x + 2y = 16 (3x4) + 2y = 16 12 + 2y = 16 2y = 16 - 12 2y = 4 y = 2 © Brain-Cells: E.Resources Ltd. All Rights Reserved 24/11/09 www.brain-cells.co.uk

Middle signs are the same so (1) - (2) will eliminate y 2. 4x + 3y = 32 (1) x + 3y = 17 (2) Middle signs are the same so (1) - (2) will eliminate y First make sure that the middle numbers are the same and both are 3y, so this is OK. Look at the signs in front of the middle numbers. Same SUBTRACT Different ADD. Finally, replace the x with 5 in one of the equations to find y It’s a good idea to label the equations (1) and (2) 3x = 15 x = 15 ÷ 3 x = 5 x + 3y = 17 5 +3y = 17 3y = 12 y = 4 © Brain-Cells: E.Resources Ltd. All Rights Reserved 24/11/09 www.brain-cells.co.uk

1. 4x + 3y = 32 x + 3y = 17 2. 3x - 2y = 17 2x + 2y = 28 3. 4x + y = 41 5x – y = 49 4. 5x - 3y = 18 3x – 3y = 6 © Brain-Cells: E.Resources Ltd. All Rights Reserved 24/11/09 www.brain-cells.co.uk

Simultaneous Equations with Different Numbers in the Middles 4x + 3y = 15 5x – 2y = 13 To solve the equations, these numbers need to be the same. The following slides explain how you can do this. © Brain-Cells: E.Resources Ltd. All Rights Reserved 24/11/09 www.brain-cells.co.uk

4x + 3y = 15 Multiply this line by 2 (2 x 4x) + (2 x 3y) = (2 x 15) (3 x 5x) – (3 x 2y) = (3 x 13) 8x + 6y = 30 15x – 6y = 39 Now, the middle numbers are the same. © Brain-Cells: E.Resources Ltd. All Rights Reserved 24/11/09 www.brain-cells.co.uk

8x + 6y = 30 15x – 6y = 39 The middles are the same If the signs are different ADD Find the value of the remaining letter 23x = 69 x = 69 ÷ 23 x = 3 8x + 6y = 30 (8 x 3) + 6y = 30 24 + 6y = 30 3y = 30 – 24 3y = 6 y = 2 © Brain-Cells: E.Resources Ltd. All Rights Reserved 24/11/09 www.brain-cells.co.uk

The middles need to be the same numbers 5x - 2y = 20 3x – 3y = 3 5x - 2y = 20 Multiply this line by 3 3x – 3y = 3 Multiply this line by 2 (3 x 5x) - (3 x 2y) = (3 x 20) (2 x 3x) – (2 x 3y) = (2 x 3) 15x - 6y = 60 6x – 6y = 6 Now, the middle numbers are the same © Brain-Cells: E.Resources Ltd. All Rights Reserved 24/11/09 www.brain-cells.co.uk

15x - 6y = 60 6x – 6y = 6 The middles are the same If the signs are the same SUBTRACT Find the value of the remaining letter 9x = 54 x = 54 ÷ 9 x = 6 3x - 3y = 3 (3 x 6) - 3y = 3 18 - 3y = 3 3y = 18 – 3 3y = 15 y = 5 © Brain-Cells: E.Resources Ltd. All Rights Reserved 24/11/09 www.brain-cells.co.uk