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Simultaneous Equations 25-Aug-14 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

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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 = 5x = 4y = = 5 x + y = 5x = 3y = = 5 x + y = 5x = 2y = = 5 x + y = 5x = 1y = = 5 © Brain-Cells: E.Resources Ltd. All Rights Reserved 24/11/09

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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 = 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

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3x + 2y = 16 (1) 2x - 2y = 4 (2) 5x = 20 x = 20 ÷ 5 x = 4 3x + 2y = 16 (3x4) + 2y = y = 16 2y = y = 4 y = 2 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 Middle signs are different so (1) + (2) will eliminate y Middle signs are different so (1) + (2) will eliminate y It’s a good idea to label the equations (1) and (2) It’s a good idea to label the equations (1) and (2) © Brain-Cells: E.Resources Ltd. All Rights Reserved 24/11/09

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2. 4x + 3y = 32 (1) x + 3y = 17 (2) 3x = 15 x = 15 ÷ 3 x = 5 x + 3y = y = 17 3y = 12 y = 4 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 Middle signs are the same so (1) - (2) will eliminate y Middle signs are the same so (1) - (2) will eliminate y It’s a good idea to label the equations (1) and (2) It’s a good idea to label the equations (1) and (2) © Brain-Cells: E.Resources Ltd. All Rights Reserved 24/11/09

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

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4x + 3y = 15 5x – 2y = 13 Simultaneous Equations with Different Numbers in the Middles 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

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

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8x + 6y = 30 15x – 6y = 39 1.The middles are the same 2.If the signs are different ADD 3.Find the value of the remaining letter 23x = 69 x = 69 ÷ 23 x = 3 8x + 6y = 30 (8 x 3) + 6y = y = 30 3y = 30 – 24 3y = 6 y = 2 © Brain-Cells: E.Resources Ltd. All Rights Reserved 24/11/09

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

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15x - 6y = 60 6x – 6y = 6 1.The middles are the same 2.If the signs are the same SUBTRACT 3.Find the value of the remaining letter 9x = 54 x = 54 ÷ 9 x = 6 3x - 3y = 3 (3 x 6) - 3y = y = 3 3y = 18 – 3 3y = 15 y = 5 © Brain-Cells: E.Resources Ltd. All Rights Reserved 24/11/09

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