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Ch 1.7 (part 5) Variable on Both Sides

Ch 1.7 (part 5) Variable on Both Sides. Objective: To solve equations where one variable exists on both sides of the equation. Rules. GOAL: Isolate the variable on one side of the equation . 1) Use the Distributive Property . (then simplify by combining LIKE Terms)

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Ch 1.7 (part 5) Variable on Both Sides

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  1. Ch 1.7 (part 5)Variable on Both Sides Objective: To solve equations where one variable exists on both sides of the equation.

  2. Rules GOAL: Isolate the variable on one side of the equation. 1) Use the Distributive Property. (then simplify by combining LIKE Terms) 2) Choose one of the variable expressions and use the Inverse Property of Addition 3) Apply the Inverse Property of Addition and/or the Inverse Property of Multiplication to the numbers. Perform Inverse operations to both sides of the equation!

  3. Special Cases x = x 2) x + 1 = x Plug in various numbers for x …….. -x -x 0 = 0 Every number makes a TRUE statement! Solution: x = All Real Numbers Plug in various numbers for x …….. -x -x 1 = 0 Every number makes a FALSE statement! Solution: x = No solution

  4. Example 1 2x+ 4 = 5x- 17 Option 2: Subtract 5x from both sides Option 1: Subtract 2x from both sides 2x + 4 = 5x - 17 2x + 4 = 5x - 17 -2x -2x -5x -5x 4 = 3x - 17 -3x + 4 =-17 +17 +17 -4 -4 21 = 3x -3x = -21 3 3 -3 - 3 7 = x x = 7

  5. Example 2 4(x - 2) - 2x = 5(x - 4) 4x - 8 - 2x = 5x - 20  Distributive Property 2x- 8 = 5x - 20  Combine LIKE Terms -2x -2x  Inverse Property of Addition for the variable -8 = 3x - 20 +20 +20  Inverse Property of Addition 12 = 3x 3 3  Inverse Property of Multiplication

  6. Example 3 Example 4 3x + 2 = 2(x - 1) + x 3x + 8 = 2(x + 4) + x 3x + 2 = 2x - 2 + x 3x + 8 = 2x + 8 + x 3x + 8 = 3x + 8 3x + 2 = 3x - 2 -3x -3x -3x -3x True ! 8 = 8 2 = -2 False ! No Solution x = any real number

  7. Classwork 1) 3x - 5 = 2x + 12 2) 5x - 3 = 13 – 3x -2x -2x +3x +3x x - 5 = 12 8x - 3 = 13 +5 +5 +3 +3 x = 17 8x = 16 8 8 x = 2 3) 2b + 6 = 7b - 9 4) -4c - 11 = 4c + 21 -2b -2b +4c +4c 6 = 5b - 9 -11 = 8c + 21 +9 +9 -21 -21 15 = 5b -32 = 8c 5 5 8 8 3 = b -4 = c

  8. 5) 3(x + 2) - (2x - 4) =- (4x + 5) 3x + 6 - 2x + 4 =- 4x - 5 x + 10 =- 4x - 5 + 4x + 4x 5x + 10 = -5 - 10 -10 5x = -15 5 5 x = -3

  9. 6) 4(y - 2) + 6y = 7(y - 8) - 3(10 - y) 4y - 8 + 6y = 7y - 56 - 30 + 3y 10y - 8 = 10y - 86 -10y -10y -8 = -86 False No Solution

  10. 7) 3(4 + k) - 2(3k + 4) = 5(k - 3) - (8k - 19) 12 + 3k - 6k - 8 = 5k - 15 - 8k + 19 -3k + 4 =-3k + 4 +3k +3k True 4 = 4 Infinitely Many Solutions! x = all real numbers

  11. 8) 5(m - 4) = 10 - 4[2(m - 5) - 5m] 5m - 20 = 10 - 4[2m - 10 - 5m] 5m - 20 = 10 - 4[-3m - 10] 5m - 20 = 10 + 12m + 40 5m - 20 = 12m + 50 -5m -5m -20 = 7m + 50 -50 -50 -70 = 7m 7 7 x = -10

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