Transcript Equations I
Revision Equations I Linear Equations By I Porter Simple Linear Equations. There are two main methods of solving linear equations: • Balanced Method • Removalist Methods The second method is a more formal solution to solving any equation. Example : Solve 2x + 4 = 18 Solution 2x + 4 2x 2x x x = 18 = 18 - 4 = 14 = 14 ÷ 2 =7 How many of these steps DO YOU have to show? That is the real question for you to address. Here is a shorter solution: 2x + 4 = 18 2x = 14 x =7 But, this is the minimum number of steps to be shown. This is the basic method the author will use for all the PowerPoints on equations. Complex Linear Equations. Solving linear equations with algebra on both sides of the equals sign. Example 1: Solve 5x + 9 = 3x - 17 Solution: 5x + 9 = 3x - 17 Remove the 3x from the RHS, by subtracting 3x from both sides. 2x + 9 = - 17 Remove the +9 from the LHS, by subtracting 9 from both sides. 2x = - 22 Remove the 2 from the LHS, by dividing both sides by 2. x = - 11 Example 2: Solve 25 - 7x = 3x - 17 Solution: 25 - 7x = 3x - 17 25 - 10x = - 17 -10x = - 42 x = 4.2 Remove the 3x from the RHS, by subtracting 3x from both sides. Remove the +25 from the LHS, by subtracting 25 from both sides. Remove the -10 from the LHS, by dividing both sides by -10. Note: There are a few different way to solve this equation. Equations with Brackets Solving linear equations with brackets. Example 1: Solve 3(2x - 1) = 4x + 17 Example 2: Solve 4(5 - 3x) = 8x - 47 Solution Solution 3(2x - 1) = 4x + 17 4(5 - 3x) = 8x - 47 6x - 3 = 4x + 17 20 - 12x = 8x -47 2x - 3 = 17 20 - 20x = -47 2x = 20 -20x = -67 x = 10 or x = 3.35 7 x=3 20 Exercise: Solve the following (write answer in fraction form) a) 5x - 17 = 3x + 35 5x - 17 2x - 17 2x x = 3x + 35 = 35 = 52 = 26 b) 3(x - 9) = 4x - 25 3(x - 9) 3x - 27 -x x = 4x - 25 = 4x - 25 =2 = -2 c) 4(2x - 9) = 18 - (x - 3) 4(2x - 9) 8x - 36 9x - 36 9x x = 18 - (x - 3) = 18 - x + 3 = 18 + 3 = 57 = 61/3 Fractional Linear Equations. To solve fractional linear equations, you need to find the L.C.M. of the denominators. Examples: Solve each of the following. 2x +1 2x - 3 7 = 4 6 3 3 (2x +1) 2 ( 2x - 3) 4 7 12 ´ -12 ´ = 12 ´ 4 6 3 a) 3(2x + 1) - 2(2x - 3) = 4(7) 6x + 3 - 4x + 6 = 28 2x + 9 = 28 2x = 19 x = 91/2 The LCM of 3, 4 and 6 is 12. Multiply each fraction by 12. Cancel the denominators and multiply by the factors remaining. Expand the brackets, take care with negative term in front of the brackets. The original equation could have been written as: 1 1 7 2x +1 2x 3 = ( ) ( ) 4 6 3 Example 3 5 = x-4 x+3 b) (x - 4)(x + 3) The LCM of (x - 4) and (x + 3) is (x - 4)(x + 3). Multiply each fraction by (x - 4)(x + 3).This is the correct mathematical method but not the fastest method. 3 5 = (x - 4)(x + 3) (x - 4) (x + 3) 3(x + 3) = 5(x - 4) Cancel common factors. Expand brackets and solve. 3x + 9 = 5x - 20 -2x + 9 = - 20 -2x = - 29 x = 141/2 This type of equation can be solved faster By cross-multiplying the denominators. 3 5 = x-4 x+3 (x + 3)´ 3 = 5 ´(x - 4) Example c) 3x - 2 3x +1 = 2x -1 2x + 3 (2x + 3)(3x - 2) = (2x -1)(3x +1) 6x 2 + 5x - 6 = 6x 2 - x -1 5x - 6 = -x -1 6x = 5 5 x= 6 Common Error In this case , cross-multiply the denominators (for speed). Expand the brackets. Cancel common factors, must be linear equations! Solve, taking care! In the following equation, the whole number must be written as a fraction with a denominator of 1. x+3 x-5 =4 2 3 x+3 x-5 4 = 2 3 1 6 ´ ( x + 3) 6 ´ ( x - 5) 6 ´ ( 4) = 2 3 1 3(x + 3)- 2(x - 5) = 24 3x + 9 - 2x +10 = 24 x = 24 - 9 -10 x=5 Exercise: Solve the following fractional equations a) 2x + 5 x + 2 = +4 3 7 b) ans : x = 5 d) 3 2 1 = 2 x-2 x+2 x - 4 ans : x = -9 e) x + 4 3- 4 x 5 - x = 2 4 8 -5 ans : x = 13 x-2 x+4 = x+3 x-5 -1 ans : x = 7 c) 2x - 5 2x = x x+3 ans : x =15 f) 3x -1 3x - 2 = 5x +1 5x + 2 ans : x = 0