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Algebra Solving Equations October 2006 ©RSH Introduction 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. October 2006 ©RSH Introduction 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. October 2006 ©RSH Notes Equations • You can ADD, SUBTRACT, MULTIPLY and DIVIDE both sides of equation by the same number. • Do the same thing to both side. October 2006 ©RSH Notes Example 1 Solve the equation x + 8 = 20 Answer x + 8 = 20 take away 8 x + 8 – 8 = 20 – 8 x = 12 Example 2 Solve the equation x - 8 = 20 Answer x - 8 = 20 x - 8 + 8 = 20 + 8 x = 28 October 2006 ©RSH add 8 Exercise Solve the following equations a) x + 5 = 17 a) x = 12 b) x – 12 = 18 b) x = 30 c) c) x + 2.5 = 8.9 October 2006 x = 6.4 ©RSH Notes You need to solve the harder equations in the Higher Tier. Harder equations • More than one step is usual. • Write down your method until you get them all right ! Example 3 Solve the equation 2x + 8 = 20 Answer 2x + 8 = 20 take away 8 2x + 8 – 8 = 20 – 8 2x = 12 2 2 x=6 October 2006 ©RSH divide by 2 Notes Example 4 Solve the equation 3x - 5 = 28 Answer 3x - 5 = 28 add 5 3x - 5 + 5 = 28 + 5 3x = 33 3 3 divide by 3 x = 11 October 2006 ©RSH Exercise Solve these equations a) b) c) d) e) f) g) h) i) j) 3x + 1 = 10 4x + 2 = 22 5x + 3 = 18 2x – 3 = 1 3x – 4 = 8 6x – 1 = 5 4x + 3 = 3 6x + 7 = 19 5x + 11 = 1 12x + 3 = 9 October 2006 a) b) c) d) e) f) g) h) i) j) 3x = 9; x = 3 4x = 20; x = 5 5x = 15; x = 3 2x = 4; x = 2 3x = 12; x = 4 6x = 6; x = 1 4x = 0; x = 0 6x = 12; x = 2 5x = -10; x = -2 12x = 6; x = ½ ©RSH Notes Equations with Brackets Expand the brackets first and then solve as usual. Example 9 2(x + 1) = 10 2x + 2 = 10 2x = 8 x=4 October 2006 Example 10 2(x - 5) = 6 2x - 10 = 6 2x = 16 x=8 ©RSH Example 11 2(x - 3)+ 2x = 6 2x - 6 + 2x = 6 4x - 6 = 6 4x = 12 x=3 Exercise Solve these equations a) b) c) d) e) 3(x + 1) = 12 4(x – 2) = 20 5(2 + x) = 25 6(2x + 1) = 12 2(x – 1) + 3x = 8 a) b) c) d) e) f) 3(x + 1) + 2(x + 2) = 14 f) g) 4(x – 2) – 2(x + 1) = 10 g) October 2006 ©RSH 3x + 3 = 12; 3x = 9; x = 3 4x – 8 = 20; 4x = 28; x = 7 10 + 5x = 25; 5x = 15; x = 3 12x + 6 = 12; 12x = 6; x = ½ 2x – 2 + 3x = 8; 5x – 2 = 8; 5x = 10; x = 2 3x + 3 + 2x + 4 = 14; 5x + 7 = 14; 5x = 7; x = 7/5 4x – 8 –2x – 2 = 10; 2x – 10 = 10; 2x = 20; x = 10 Notes Harder again • Don’t let the fractions get to you ! Example 5 Solve the equation 3x = 2 Answer 3x 2 3x 2 3 3 2 x 3 It’s fine to Leave your Answer as a Fraction. October 2006 ©RSH divide by 3 Notes Harder again • Don’t let the fractions get to you ! Example 6 Solve the equation October 2006 x 6 5 Answer x 6 5 x 5 65 5 x 30 ©RSH multiply by 5 Notes Harder again • Don’t let the fractions get to you ! Example 7 Solve the equation October 2006 2x 4 7 Answer 2x 7 2x 7 7 2x 2x 2 x ©RSH 4 47 28 28 2 14 Exercise Solve these equations a) b) c) d) e) October 2006 x 7 5 x 13 10 3x 9 2 2x 3 5 5x 6 4 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 ©RSH Notes Harder again • Step by step Answer Example 8 2x 2 10 Solve the equation 7 October 2006 2x 2 10 7 2x 22 7 2x 7 2x 7 7 2x 2x 2 x ©RSH 10 2 8 87 56 56 2 28 Exercise Solve these equations a) b) c) d) e) October 2006 x 1 7 5 x 7 12 2 x 7 2 3 2x 5 11 3 5x 2 13 3 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 ©RSH Notes …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. Example 12 Solve 3x + 1 = x + 3 Answer 3x + 1 = x + 3 3x + 1 – 1 = x + 3 – 1 3x = x + 2 3x – x = x – x + 2 2x = 2 x=1 October 2006 ©RSH subtract 1 subtract x divide by 2 Exercise Solve these equations a) b) c) d) e) 3x + 7 = x – 3 5x – 5 = 7 – x 2x – 7 = 5 – 3x 2x + 4 = x – 3 7 – 2x = 2x - 7 October 2006 a) b) c) d) e) ©RSH 3x = x – 10; 2x = -10; x = 5 6x = 12; x = 2 5x = 12; x = 12/5 or 2.4 x = -7 14 = 4x; 4x = 14; x = 3.5