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Topic: EQUATIONS Simple Equations Fractional Equations Guidelines • Equations must be balanced. You must respect the laws of equations. • The goals is to bring variable on the left and number to the right. • Coefficient (or number) of variable +1. For example, 1x = 5; just put x = 5 Examples 1. #1) Solve: x = 5 + 2 x=7 2. x = 5 (2 + 7) – ( 7 – 3) x = 10 + 35 – 7 + 3 • Remember a minus before a bracket changes the sign of everything in the bracket x = 41 Guidelines con’t • Whatever you do to one side, you must do to another. If you add 5 to one side, you must add 5 to the other side. • If you have a number near the variable, always divide it by that number. For example 2x = 10; divide both by 2. x = 5 • If -5x = 10; divide by -5; x = -2 • When it changes signs, it changes signs Example • 4x + 1 = 13 4x = 13 - 1 (the 1 changed sides so it changes signs) 4x = 12 (divide both sides by 4) x=3 Verification • If you want to guarantee that you have the right answer you should verify. To verify: replace the number into the letter in the question. • 4x + 1 = 13 (Original question) & x=3 4 (3) + 1 = 13 12+1=13 13=13 This is true; you have the right answer. Guideline & Example • • • • 3x – 5 = 10x + 10 -3x - 10x = 10 + 5 -13x = 15 x = -1.15 Another Example – Long Version • • • • 3x – 5 = 8x + 15 3x – 8x – 5 + 5 = 8x – 8x + 15 + 5 -5x = 20 x = -4 Same example – Short Version • • • • 3x – 5 = 8x + 15 -5x = 20 x = -4 We will continue with the short version ;) More examples • • • • • • • 4x + 7 = 2x – 11 2x = -18 x=-9 Verify! 4 (-9) + 7 = 2 (-9) – 11 -36 + 7 = -18 – 11 - 29 = -29 (You have the right answer) Another Example • 9x – 5 = 2x + 4 7x = 9 x = 1.29 You can still verify this! 9 (1.29) – 5 = 2 (1.29) + 4 6.61 = 6.58 (rounding error… close enough!) More Examples • 3x + 5 = 6x + 25 • x = - 6.67 • 4x + 2 = 8x – 31 • X = 8.25 --- are you verifying?) Reminders • Distributive property • 5(3x + 2) means you multiply 5 by everything in the bracket • 15x + 10 • A minus sign before the bracket changes the sign of everything in the bracket Now to add some fun – and have them longer • 3 (5x-7) – (2x+8) = 2 (3x-1) • 15x – 21 – 2x – 8 = 6x – 2 • CLEAN IT UP BEFORE MOVING NUMBERS OR LETTERS • 13x – 29 = 6x – 2 • 7x = 27 • x = 3.86 • Verify! Verify • 3 (5x-7) – (2x+8) = 2 (3x-1) • 3 (5(3.86) – 7) – (2(3.86) - 8 = 2 (3(3.86) – 1) • 3 (12.3) – 7.72 – 8 = 2 (10.58) • 21.18 = 21.16 (rounding error – close enough!) Quiz # 3 Equations • 1) • 2) • 3) • 4) • 5) 5x 2x 9x 3x 6x – – – – – 7 5 2 7 1 = = = = = 3x + 7 4x + 7 - 40 + 5x 8x + 20 8x + 20 Quiz #3 Con’t • 6) 2 (3x – 7) = 5 (3x – 1) • 7) 7(4x + 1) – 5 (3x + 5) = 8x – (3x + 2) • 8) 2x – (3x + 1) • 9) 2x + 5 • 10) – 5 – 5 Quiz #3 Equation Solutions • • • • • • • • • • 1) 7 (1a) 2) – 6 (1b) 3) -9.5 (1i) 4) – 5.4 (1o) 5) -10.5 6) -1 (1u) 7) 2 (1w) 8) –x – 1 (minus before a bracket!) 9) 2x + 5 (don’t mix apples & oranges!) 10) – 10 Fractional Equations • Once again we want to get rid of the fraction • Find the LCD (Lowest Common Denominator) • Multiply every term to get the LCD. Example 1. x – 2 = 11 5 3 15 LCD of 5, 3 and 15 is LCD is 15 3x – 10 = 11 3x = 21 X = 7 and then yes… VERIFY Verify 1. x – 2 = 11 5 3 15 7/5 – 2/3 = 11 / 15 0.73 = 0.73 it works! Another example • 2x – 11 = 3x – 5 7 14 28 7 8x – 22 = 3x – 20 5x = 2 x = 0.4 Another Example • 3 – 11x = 5 + 5x 8 12 24 6 9 – 22x = 5 + 20x -42x = -4 X = 0.1 Final Example (Hard) • 3x – 2 – 11x + 8x = 3 – 7x + 1 5 3 30 15 2 15 30 18x – 20 – 11x + 16x = 45 – 14x +1 37x = 66 X = 1.78