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Unit 3 Lecture 25 Solving Equations with Algebraic Fractions Solving Equations with Fractions Lecture 25 1 Objectives • To solve an equation with algebraic fractions Lecture 25 2 3x 7 x 5 2 10 Lecture 25 Solve the equation 3 3x 7 x 5 2 10 Solve the equation Multiply both sides by 10 Lecture 25 4 3x 7 x 5 2 10 (10)3 x (10)7 (10)x 5 2 10 Lecture 25 Solve the equation Multiply both sides by 10 5 3x 7 x 5 2 10 (10)3 x (10)7 (10)x 5 2 10 Solve the equation Multiply both sides by 10 Eliminate the denominators Lecture 25 6 3x 7 x 5 2 10 (10)3 x (10)7 (10)x 5 2 10 2 5 (10) 3 x 5 (10) 7 2 (10) x Solve the equation Multiply both sides by 10 Eliminate the denominators 10 Lecture 25 7 3x 7 x 5 2 10 (10)3 x (10)7 (10)x 5 2 10 2 5 (10) 3 x 5 (10) 7 2 6x + 35 (10) x Solve the equation Multiply both sides by 10 Eliminate the denominators 10 = x Lecture 25 8 3x 7 x 5 2 10 (10)3 x (10)7 (10)x 5 2 10 2 5 (10) 3 x 5 (10) 7 2 6x + 35 (10) x Solve the equation Multiply both sides by 10 Eliminate the denominators 10 = x Lecture 25 Add –6x to both sides 9 3x 7 x 5 2 10 (10)3 x (10)7 (10)x 5 2 10 2 5 (10) 3 x 5 (10) 7 2 (10) x Solve the equation Multiply both sides by 10 Eliminate the denominators 10 6x + 35 = x 35 = -5x Lecture 25 Add –6x to both sides 10 3x 7 x 5 2 10 (10)3 x (10)7 (10)x 5 2 10 2 5 (10) 3 x 5 (10) 7 2 (10) x Solve the equation Multiply both sides by 10 Eliminate the denominators 10 6x + 35 = x 35 = -5x Lecture 25 Add –6x to both sides Divide both sides by -5 11 3x 7 x 5 2 10 (10)3 x (10)7 (10)x 5 2 10 2 5 (10) 3 x 5 (10) 7 2 (10) x Solve the equation Multiply both sides by 10 Eliminate the denominators 10 6x + 35 = x 35 = -5x -7 = x Lecture 25 Add –6x to both sides Divide both sides by -5 Solution 12 4 4 8 x 3 5x Solve the equation Lecture 25 13 4 4 8 x 3 5x Solve the equation Multiply both sides by 15x Lecture 25 14 4 4 8 x 3 5x Solve the equation (15 x )4 (15 x )4 (15 x )8 x 3 5x Lecture 25 Multiply both sides by 15x 15 4 4 8 x 3 5x Solve the equation (15 x )4 (15 x )4 (15 x )8 Multiply both sides by 15x x 3 5x Eliminate the denominators Lecture 25 16 4 4 8 x 3 5x Solve the equation (15 x )4 (15 x )4 (15 x )8 x 3 5x 5 3 (15 x )4 ( 15 x )4 ( 15 x )8 x 3 5x Lecture 25 Multiply both sides by 15x Eliminate the denominators 17 4 4 8 x 3 5x Solve the equation (15 x )4 (15 x )4 (15 x )8 x 3 5x 5 3 (15 x )4 ( 15 x )4 ( 15 x )8 x 3 5x 60 - 20x Multiply both sides by 15x Eliminate the denominators = 24 Lecture 25 18 4 4 8 x 3 5x (15 x )4 (15 x )4 (15 x )8 x 3 5x 5 3 (15 x )4 ( 15 x )4 ( 15 x )8 x 3 5x 60 - 20x = 24 Lecture 25 Solve the equation Multiply both sides by 15x Eliminate the denominators Add –60 to both sides 19 4 4 8 x 3 5x Solve the equation (15 x )4 (15 x )4 (15 x )8 Multiply both sides by 15x x 3 5x 5 3 Eliminate the x )8 denominators (15 x )4 ( 15 x )4 ( 15 x 3 5x 60 - 20x = 24 -20x = -36 Lecture 25 Add –60 to both sides 20 4 4 8 x 3 5x (15 x )4 (15 x )4 (15 x )8 x 3 5x 5 3 (15 x )4 ( 15 x )4 ( 15 x )8 x 3 5x 60 - 20x = 24 -20x = -36 Lecture 25 Solve the equation Multiply both sides by 15x Eliminate the denominators Add –60 to both sides Divide both sides by -20 21 4 4 8 x 3 5x (15 x )4 (15 x )4 (15 x )8 x 3 5x 5 3 (15 x )4 ( 15 x )4 ( 15 x )8 x 3 5x 60 - 20x = 24 -20x = -36 x = 1.8 Lecture 25 Solve the equation Multiply both sides by 15x Eliminate the denominators Add –60 to both sides Divide both sides by -20 Solution 22 2x x 2 5 6 Lecture 25 Solve the equation 23 2x x 2 5 6 Solve the equation Multiply both sides by 30 Lecture 25 24 2x x 2 5 6 (30)2 x (30)( x 2) 5 6 Lecture 25 Solve the equation Multiply both sides by 30 25 2x x 2 5 6 (30)2 x (30)( x 2) 5 6 6 5 (30) 2 x (30) ( x 2) 5 6 Lecture 25 Solve the equation Multiply both sides by 30 26 2x x 2 5 6 (30)2 x (30)( x 2) 5 6 6 5 (30) 2 x (30) ( x 2) 5 6 Lecture 25 Solve the equation Multiply both sides by 30 Eliminate the denominators 27 2x x 2 5 6 (30)2 x (30)( x 2) 5 6 6 5 (30) 2 x (30) ( x 2) 5 6 12x = 5(x+2) Lecture 25 Solve the equation Multiply both sides by 30 Eliminate the denominators 28 2x x 2 5 6 (30)2 x (30)( x 2) 5 6 6 5 (30) 2 x (30) ( x 2) 5 6 12x = 5(x+2) Lecture 25 Solve the equation Multiply both sides by 30 Eliminate the denominators Distributive Property 29 2x x 2 5 6 (30)2 x (30)( x 2) 5 6 6 5 (30) 2 x (30) ( x 2) 5 6 12x = 5(x+2) Solve the equation Multiply both sides by 30 Eliminate the denominators Distributive Property 12x = 5x+10 Lecture 25 30 2x x 2 5 6 (30)2 x (30)( x 2) 5 6 6 5 (30) 2 x (30) ( x 2) 5 6 12x = 5(x+2) 12x = 5x+10 Lecture 25 Solve the equation Multiply both sides by 30 Eliminate the denominators Distributive Property Add –5x to both sides 31 2x x 2 5 6 (30)2 x (30)( x 2) 5 6 6 5 (30) 2 x (30) ( x 2) 5 6 12x = 5(x+2) 12x = 5x+10 Solve the equation Multiply both sides by 30 Eliminate the denominators Distributive Property Add –5x to both sides 7x = 10 Lecture 25 32 2x x 2 5 6 (30)2 x (30)( x 2) 5 6 6 5 (30) 2 x (30) ( x 2) 5 6 12x = 5(x+2) 12x = 5x+10 Solve the equation Multiply both sides by 30 Eliminate the denominators Distributive Property Add –5x to both sides Divide both sides by 7 7x = 10 Lecture 25 33 2x x 2 5 6 (30)2 x (30)( x 2) 5 6 6 5 (30) 2 x (30) ( x 2) 5 6 12x = 5(x+2) Solve the equation Multiply both sides by 30 Eliminate the denominators Distributive Property 12x = 5x+10 Add –5x to both sides 7x = 10 x = 1.43 Divide both sides by 7 Solution Lecture 25 34 y 4 y 3 5 2 9 18 Lecture 25 Solve the equation 35 y 4 y 3 5 2 9 18 Solve the equation Multiply both sides by 18 Lecture 25 36 y 4 y 3 5 Solve the equation 2 9 18 (18)( y 4) (18)( y 3) (18)5 Multiply both sides by 18 2 9 18 Lecture 25 37 y 4 y 3 5 Solve the equation 2 9 18 (18)( y 4) (18)( y 3) (18)5 Multiply both sides by 18 2 9 18 Eliminate the denominators Lecture 25 38 y 4 y 3 5 Solve the equation 2 9 18 (18)( y 4) (18)( y 3) (18)5 Multiply both sides by 18 2 9 1 8 9 2 (18) ( y 4) (18) ( y 3) (18) 5 Eliminate the denominators 2 9 18 Lecture 25 39 y 4 y 3 5 Solve the equation 2 9 18 (18)( y 4) (18)( y 3) (18)5 Multiply both sides by 30 2 9 1 8 9 2 (18) ( y 4) (18) ( y 3) (18) 5 Eliminate the denominators 2 9 18 9 (y-4) – 2 (y-3) = 5 Lecture 25 40 y 4 y 3 5 Solve the equation 2 9 18 (18)( y 4) (18)( y 3) (18)5 Multiply both sides by 30 2 9 1 8 9 2 (18) ( y 4) (18) ( y 3) (18) 5 Eliminate the denominators 2 9 18 9 (y-4) – 2 (y-3) = 5 Lecture 25 Distributive Property 41 y 4 y 3 5 Solve the equation 2 9 18 (18)( y 4) (18)( y 3) (18)5 Multiply both sides by 30 2 9 1 8 9 2 (18) ( y 4) (18) ( y 3) (18) 5 Eliminate the denominators 2 9 18 9 (y-4) – 2 (y-3) = 5 Distributive Property 9y – 36 - 2y + 6 = 5 Lecture 25 42 y 4 y 3 5 Solve the equation 2 9 18 (18)( y 4) (18)( y 3) (18)5 Multiply both sides by 30 2 9 1 8 9 2 (18) ( y 4) (18) ( y 3) (18) 5 Eliminate the denominators 2 9 18 9 (y-4) – 2 (y-3) = 5 Distributive Property 9y – 36 - 2y + 6 = 5 Combine like terms Lecture 25 43 y 4 y 3 5 Solve the equation 2 9 18 (18)( y 4) (18)( y 3) (18)5 Multiply both sides by 18 2 9 1 8 9 2 (18) ( y 4) (18) ( y 3) (18) 5 Eliminate the denominators 2 9 18 9 (y-4) – 2 (y-3) = 5 Distributive Property 9y – 36 - 2y + 6 = 5 Combine like terms 7y - 30 = 5 Lecture 25 44 y 4 y 3 5 Solve the equation 2 9 18 (18)( y 4) (18)( y 3) (18)5 Multiply both sides by 18 2 9 1 8 9 2 (18) ( y 4) (18) ( y 3) (18) 5 Eliminate the denominators 2 9 18 9 (y-4) – 2 (y-3) = 5 Distributive Property 9y – 36 - 2y + 6 = 5 Combine like terms 7y - 30 = 5 Lecture 25 Add 30 to both sides 45 y 4 y 3 5 Solve the equation 2 9 18 (18)( y 4) (18)( y 3) (18)5 Multiply both sides by 18 2 9 1 8 9 2 (18) ( y 4) (18) ( y 3) (18) 5 Eliminate the denominators 2 9 18 9 (y-4) – 2 (y-3) = 5 Distributive Property 9y – 36 - 2y + 6 = 5 Combine like terms 7y - 30 = 5 7y = 35 Lecture 25 Add 30 to both sides 46 y 4 y 3 5 Solve the equation 2 9 18 (18)( y 4) (18)( y 3) (18)5 Multiply both sides by 18 2 9 1 8 9 2 (18) ( y 4) (18) ( y 3) (18) 5 Eliminate the denominators 2 9 18 9 (y-4) – 2 (y-3) = 5 Distributive Property 9y – 36 - 2y + 6 = 5 Combine like terms 7y - 30 = 5 7y = 35 Lecture 25 Add 30 to both sides Divide both sides by 7 47 y 4 y 3 5 Solve the equation 2 9 18 (18)( y 4) (18)( y 3) (18)5 Multiply both sides by 18 2 9 1 8 9 2 (18) ( y 4) (18) ( y 3) (18) 5 Eliminate the denominators 2 9 18 9(y-4) – 2(y – 3) = 5 Distributive Property 9y – 36 – 2y + 6 = 5 Combine like terms 7y - 30 = 5 7y = 35 y=5 Lecture 25 Add 30 to both sides Divide both sides by 7 Solution 48