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Solving Two-Step Equations

Solving Two-Step Equations. Section 2-2. Goals. Goal. Rubric. Level 1 – Know the goals. Level 2 – Fully understand the goals. Level 3 – Use the goals to solve simple problems. Level 4 – Use the goals to solve more advanced problems .

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Solving Two-Step Equations

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  1. Solving Two-Step Equations Section 2-2

  2. Goals Goal Rubric Level 1 – Know the goals. Level 2 – Fully understand the goals. Level 3 – Use the goals to solve simple problems. Level 4 – Use the goals to solve more advanced problems. Level 5 – Adapts and applies the goals to different and more complex problems. • To solve two-step equations in one variable.

  3. Vocabulary • None

  4. Operations in the equation To solve   Subtract 5 from both First x is multiplied by 2. sides of the equation.  Then 5 is added.  Then divide both sides by 2. Solving Two-Step Equations Many equations contain more than one operation, such as 2x + 5 = 11. This equation contains multiplication and addition. Equations that contain two operations require two steps to solve. Identify the operations in the equation and the order in which they are applied to the variable. Then use inverse operations to undo them in reverse over one at a time. 2x + 5 = 11

  5. –5 –5 2x = 6 2x + 5 =11 Subtract 5 from both sides of the equation. Divide both sides of the equation by 2. x = 3 The solution set is {3}. Each time you perform an inverse operation, you create an equation that is equivalentto the original equation. Equivalent equations have the same solutions, or the same solution set. In the example above, 2x + 5 = 11, 2x = 6, and x = 3 are all equivalent equations.

  6. Ex: x + 9 = 6 5 Ask yourself: What is the first thing we are doing to x? The second thing? Recall the order of operations as you answer these questions. dividing by 5 adding 9 To undo these steps, do the inverse operations in inverse order. To Solve: Inverse Operations in the Inverse Order

  7. Use a chart as a shortcut to answering the questions. DO UNDO ÷5 -9 +9 ·5 Follow the steps in the ‘undo’ column to isolate the variable. Ex: x + 9 = 6 5 First subtract 9. x + 9 - 9 = 6 - 9 5 x = -3 5 Then multiply by 5. (5) x = -3(5) 5 x = -15 The DO-UNDO Chart

  8. Complete the do-undo chart. DO UNDO -2 ·3 ÷ 3 +2 To solve for d: First multiply by 3. Then add 2. Ex: d - 2 = 7 3 (3) d - 2 = 7(3) 3 d - 2 = 21 d - 2 = 21 +2 +2 d = 23 Example:

  9. Remember to always use the sign in front of the number. DO UNDO ÷ -7 - 3 +3 · -7 To solve for a: First subtract 3. Then multiply by -7. Ex: 3 - a = -2 7 3 - a = -2 7 -3 -3 - a = -5 7 (-7)(- a) = (-5)(-7) 7 a = 35 Example:

  10. Your Turn: • 5z + 16 = 51 • 14n - 8 = 34 • 4b + 8 = 10 -2

  11. DO UNDO · 5 - 16 +16 ÷ 5 z = 7 DO UNDO · 14 +8 -8 ÷ 14 n = 3 The answers: DO UNDO · 4 · -2 +8 - 8 ÷ -2 ÷ 4 b = -7

  12. + 4 +4 Your Turn: Solve the equation. Check your answer. –4+ 7x = 3 First x is multiplied by 7. Then –4 is added. –4+ 7x = 3 Add 4 to both sides. 7x = 7 7x = 7 is equivalent to –4 + 7x = 3. Since x is multiplied by 7, divide both sides by 7 to undo the multiplication. x = 1 The solution set is {1}.

  13. Your Turn: Continued Check your answer. Check –4+ 7x = 3 To check your solution, substitute 1 for x in the original equation. –4 + 7(1) 3 –4 + 7 3 33 

  14. + 5.7+5.7 Your Turn: Solve the equation. 1.5 = 1.2y – 5.7 First y is multiplied by 1.2. Then 5.7 is subtracted. 1.5 = 1.2y – 5.7 Add 5.7 to both sides. 7.2 = 1.2y 7.2 = 1.2y is equivalent to 1.5 = 1.2y – 5.7. Since y is multiplied by 1.2, divide both sides by 1.2 to undo the multiplication. 6= y The solution set is {6}.

  15. –2 –2 = 0 is equivalent to + 2 = 2. = 0 Your Turn: Solve the equation. First n is divided by 7. Then 2 is added. Subtract 2 from each side. Since n is divided by 7, multiply both sides by 7 to undo the division. n = 0 The solution set is {0}.

  16. Since is subtracted from , add to both sides to undo the subtraction. Example: Two-Step Equations with Fractions Solve the equation. Method 1 Use fraction operations.

  17. Example: Continued Since y is divided by 8 multiply both sides by 8. Simplify. y = 16 The solution set is {16}.

  18. +6 +6 Example: Continued Method 2 Multiply by the least common denominator (LCD) to clear fractions. Multiply both sides by 8, the LCD of the fractions. Distribute 8 on the left side. Simplify. Since 6 is subtracted from y, add 6 to both sides to undo the subtraction. y – 6 = 10 y = 16 The solution set is {16}.

  19. Since is added to , subtract from both sides to undo the addition. Example: Two-Step Equations with Fractions Solve the equation. Method 1 Use fraction operations.

  20. Since r is multiplied by multiply both sides by , the reciprocal. The solution set is . Example: Continued Simplify.

  21. Example: Continued Method 2 Multiply by the least common denominator (LCD) to clear the fractions. Multiply both sides by 12, the LCD of the fractions. Distribute 12 on the left side.

  22. – 9 –9 8r =–2 The solution set is . Example: Continued 8r + 9 = 7 Simplify. Since 9 is added 8r, subtract 9 from both sides to undo the addition. Since r is multiplied by 8, divide both sides 8 to undo the multiplication.

  23. Helpful Hint You can multiply both sides of the equation by any common denominator of the fractions. Using the LCD is the most efficient.

  24. Since is subtracted from , add to both sides to undo the subtraction. Your Turn: Solve the equation. Check your answer. Method 1 Use fraction operations.

  25. Since x is multiplied by multiply both sides by , the reciprocal. The solution set is . Your Turn: Continued Simplify.

  26. Your Turn: Continued Method 2 Multiply by the least common denominator (LCD) to clear the fractions. Multiply both sides by 10, the LCD of the fractions. Distribute 10 on the left side. 4x – 5 = 50

  27. + 5 +5 4x = 55 The solution set is . Your Turn: Continued 4x – 5 = 50 Simplify. Since 5 is subtracted from 4x add 5 to both sides to undo the subtraction. Simplify. Since x is multiplied by 4, divide both sides 4 to undo the multiplication.

  28. Since is added to , subtract from both sides to undo the addition. Your Turn: Solve the equation. Method 1 Use fraction operations.

  29. Since u is multiplied by multiply both sides by the reciprocal, . The solution set is . Your Turn: Continued Simplify.

  30. Your Turn: Continued Method 2 Multiply by the least common denominator (LCD) to clear fractions. Multiply both sides by 8, the LCD of the fractions. Distribute 8 on the left side. 6u + 4 = 7

  31. – 4 – 4 6u = 3 The solution set is . Your Turn: Continued 6u + 4 = 7 Simplify. Since 4 is added to 6u subtract 4 from both sides to undo the addition. Simplify. Since u is multiplied by 6, divide both sides 6 to undo the multiplication.

  32. Since is subtracted from , add to both sides to undo the subtraction. Your Turn: Solve the equation. Method 1 Use fraction operations.

  33. Since n is multiplied by multiply both sides by the reciprocal, . Your Turn: Continued Simplify. The solution set is {15}. n = 15

  34. Joke Time • Who made King Arthur’s round table? • Sir - Cumference. • Where was the Declaration of Independence signed? • At the bottom. • What was Camelot famous for? • It’s knight life!

  35. Assignment • 2.2 Exercises Pg. 98 – 100: #10 – 68 even

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