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October 2006 ©RSH Algebra Solving Equations

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October 2006 ©RSH Equations Many people use equations (sometimes without knowing !) A plumber charges £35 for a call out then £17 per hour. If the bill is £69, how long did the job take ? 17x + 35 = 69 That’s the equation. Equations Many people use equations (sometimes without knowing !) A plumber charges £35 for a call out then £17 per hour. If the bill is £69, how long did the job take ? 17x + 35 = 69 That’s the equation. Introduction

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October 2006 ©RSH Equations Solving equations means finding the value of the unknown (often written as ‘x’). Some equations are easy to solve by guessing x + 7 = 9 x = 2 Others are not. A set of easy to remember rules will help you to solve ANY equation. Equations Solving equations means finding the value of the unknown (often written as ‘x’). Some equations are easy to solve by guessing x + 7 = 9 x = 2 Others are not. A set of easy to remember rules will help you to solve ANY equation. Introduction

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October 2006 ©RSH Equations You can ADD, SUBTRACT, MULTIPLY and DIVIDE both sides of equation by the same number. Do the same thing to both side. Equations You can ADD, SUBTRACT, MULTIPLY and DIVIDE both sides of equation by the same number. Do the same thing to both side. Notes

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October 2006 ©RSH Example 1 Solve the equation x + 8 = 20 Example 1 Solve the equation x + 8 = 20 Notes Answer x + 8 = 20take away 8 x + 8 – 8 = 20 – 8 x = 12 Answer x + 8 = 20take away 8 x + 8 – 8 = 20 – 8 x = 12 Example 2 Solve the equation x - 8 = 20 Example 2 Solve the equation x - 8 = 20 Answer x - 8 = 20add 8 x = x = 28 Answer x - 8 = 20add 8 x = x = 28

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October 2006 ©RSH Exercise Solve the following equations a)x = 12 b)x = 30 c)x = 6.4 a)x + 5 = 17 b)x – 12 = 18 c)x = 8.9

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October 2006 ©RSH Harder equations More than one step is usual. Write down your method until you get them all right ! Harder equations More than one step is usual. Write down your method until you get them all right ! Notes You need to solve the harder equations in the Higher Tier. Example 3 Solve the equation 2x + 8 = 20 Example 3 Solve the equation 2x + 8 = 20 Answer 2x + 8 = 20take away 8 2x + 8 – 8 = 20 – 8 2x = 12divide by 2 x = 6 Answer 2x + 8 = 20take away 8 2x + 8 – 8 = 20 – 8 2x = 12divide by 2 x = 6 22

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October 2006 ©RSH Notes Example 4 Solve the equation 3x - 5 = 28 Example 4 Solve the equation 3x - 5 = 28 Answer 3x - 5 = 28add 5 3x = x = 33divide by 3 x = 11 Answer 3x - 5 = 28add 5 3x = x = 33divide by 3 x = 11 33

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October 2006 ©RSH Exercise Solve these equations a)3x = 9; x = 3 b)4x = 20; x = 5 c)5x = 15; x = 3 d)2x = 4; x = 2 e)3x = 12; x = 4 f)6x = 6; x = 1 g)4x = 0; x = 0 h)6x = 12; x = 2 i)5x = -10; x = -2 j)12x = 6; x = ½ a)3x + 1 = 10 b)4x + 2 = 22 c)5x + 3 = 18 d)2x – 3 = 1 e)3x – 4 = 8 f)6x – 1 = 5 g)4x + 3 = 3 h)6x + 7 = 19 i)5x + 11 = 1 j)12x + 3 = 9

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October 2006 ©RSH Equations with Brackets Expand the brackets first and then solve as usual. Equations with Brackets Expand the brackets first and then solve as usual. Notes Example 9 2(x + 1) = 10 2x + 2 = 10 2x = 8 x = 4 Example 9 2(x + 1) = 10 2x + 2 = 10 2x = 8 x = 4 Example 10 2(x - 5) = 6 2x - 10 = 6 2x = 16 x = 8 Example 10 2(x - 5) = 6 2x - 10 = 6 2x = 16 x = 8 Example 11 2(x - 3)+ 2x = 6 2x x = 6 4x - 6 = 6 4x = 12 x = 3 Example 11 2(x - 3)+ 2x = 6 2x x = 6 4x - 6 = 6 4x = 12 x = 3

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October 2006 ©RSH Exercise Solve these equations a)3x + 3 = 12; 3x = 9; x = 3 b)4x – 8 = 20; 4x = 28; x = 7 c)10 + 5x = 25; 5x = 15; x = 3 d)12x + 6 = 12; 12x = 6; x = ½ e)2x – 2 + 3x = 8; 5x – 2 = 8; 5x = 10; x = 2 f)3x x + 4 = 14; 5x + 7 = 14; 5x = 7; x = 7/5 g)4x – 8 –2x – 2 = 10; 2x – 10 = 10; 2x = 20; x = 10 a)3(x + 1) = 12 b)4(x – 2) = 20 c)5(2 + x) = 25 d)6(2x + 1) = 12 e)2(x – 1) + 3x = 8 f)3(x + 1) + 2(x + 2) = 14 g)4(x – 2) – 2(x + 1) = 10

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October 2006 ©RSH Harder again Don’t let the fractions get to you ! Harder again Don’t let the fractions get to you ! Notes Example 5 Solve the equation 3x = 2 Example 5 Solve the equation 3x = 2 Answer divide by 3 Answer divide by 3 It’s fine to Leave your Answer as a Fraction.

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October 2006 ©RSH Harder again Don’t let the fractions get to you ! Harder again Don’t let the fractions get to you ! Notes Example 6 Solve the equation Example 6 Solve the equation Answer multiply by 5 Answer multiply by 5

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October 2006 ©RSH Harder again Don’t let the fractions get to you ! Harder again Don’t let the fractions get to you ! Notes Example 7 Solve the equation Example 7 Solve the equation Answer

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October 2006 ©RSH Exercise Solve these equations a)x = 35 b)x = 130 c)3x = 18; x = 6 d)2x = 15; x = 7½ e)5x = -24; x = -24/5 or –4.8

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October 2006 ©RSH Harder again Step by step Harder again Step by step Notes Example 8 Solve the equation Example 8 Solve the equation Answer

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October 2006 ©RSH Exercise Solve these equations a)x/5 = 6; x = 30 b)x/2 = 5; x = 10 c)x/3 = 9; x = 27 d)2x/3 = 6; 2x = 18; x = 9 e)5x/3 = 15; 5x = 45; x = 9

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October 2006 ©RSH …and finally What happens if there is an x on both sides. Re-arrange to get all the x’s on one side and numbers on the other. …and finally What happens if there is an x on both sides. Re-arrange to get all the x’s on one side and numbers on the other. Notes Example 12 Solve3x + 1 = x + 3 Answer 3x + 1 = x + 3subtract 1 3x + 1 – 1 = x + 3 – 1 3x = x + 2 3x – x = x – x + 2subtract x 2x = 2divide by 2 x = 1 Example 12 Solve3x + 1 = x + 3 Answer 3x + 1 = x + 3subtract 1 3x + 1 – 1 = x + 3 – 1 3x = x + 2 3x – x = x – x + 2subtract x 2x = 2divide by 2 x = 1

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October 2006 ©RSH Exercise Solve these equations a)3x = x – 10; 2x = -10; x = 5 b)6x = 12; x = 2 c)5x = 12; x = 12/5 or 2.4 d)x = -7 e)14 = 4x; 4x = 14; x = 3.5 a)3x + 7 = x – 3 b)5x – 5 = 7 – x c)2x – 7 = 5 – 3x d)2x + 4 = x – 3 e)7 – 2x = 2x - 7

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