1. CONFIDENTIAL 1 Solving Two-Step and Multi-Step Equations

2. CONFIDENTIAL 2 Warm Up Simplify: 1) K = 8 4 2) z = -9 3 3) -2 = w -7 4) 6 = t -5 1) k = 32 2) z = -27 3) w = 14 4) t = - 30

3. CONFIDENTIAL 3 Introduction Equations containing more than one operation can model real-world situations, such as the cost of a music club membership. Alex belongs to a music club. In this club, students can buy a student discount card for $19.95. This card allows them to buy CDs for $3.95 each. After one year, Alex has spent $63.40. To find the number of CDs c that Alex bought, you can solve an equation. $3.95c + $19.95 = $63.40 Cost of discount card Cost per CDTotal cost Next page

4. CONFIDENTIAL 4 Notice that this equation contains multiplication and addition. Equations that contain more than one operation require more than one step to solve. Identify the operations in the equation and the order in which they are applied to the variable. Then use inverse operations and work backward to undo them one at a time. $3.95c + $19.95 = $63.40 Operations in the EquationTo Solve 1) First c is multiplied by 3.95. 2) Then 19.95 is added. 1) Subtract 19.95 from both sides of the equation. 2) Then divide both sides by 3.95.

5. CONFIDENTIAL 5 Solving Two-Step Equations Solve 10 = 6 - 2x. Check your answer. 10 = 6 - 2x First x is multiplied by -2. Then 6 is added. - 6 -6 Work backward: Subtract 6 from both sides. 4 = -2x -2 -2 = x Since x is multiplied by -2, divide both sides by -2 to undo the multiplication. Check 10 = 6 - 2x 10 6 - 2 (-2) 6 - (-4) 10

6. CONFIDENTIAL 6 Solve each equation. Check your answer. Now you try! 1) -4 + 7x = 3 2) 1.5 = 1.2 y - 5.7 3) n + 2 = 2 7 1) x = 1 2) y = 6 3) n = 0

7. CONFIDENTIAL 7 Solving Two-Step Equations That Contain Fractions Solve: q – 1 = 3 15 5 5 Method 1: Use fraction operations. q – 1 = 3 15 5 5 Since 1/5 is subtracted from q/5 15, add 1/5 to both sides to undo the subtraction. – 1 - 1 5 5 q = 4 15 5 15 × q = 15 × 4 = 60 15 5 5 Since q is divided by 15, multiply both sides by 15 to undo the division. Next page Simplify. q = 12

8. CONFIDENTIAL 8 Method 2: Multiply by the least common denominator (LCD) to clear the fractions. q – 1 = 3 15 5 5 Multiply both sides by 15, the LCD of the fractions. q – 1 = 3 15 5 5 15 × q – 1 = 3 15 5 5 15 × Distribute 15 on the left side. q - 3 = 9 + 3 + 3 q = 12 Simplify Since 3 is subtracted from q, add 3 to both sides to undo the subtraction.

9. CONFIDENTIAL 9 Solve each equation. Check your answer. Now you try! 1) 2x – 1 = 5 5 2 2) 3u + 1 = 7 4 2 8 3) n - 1 = 8 5 3 3 3) n = 15 1)x = 55 4 2) u = 1 2

10. CONFIDENTIAL 10 Simplifying Before Solving Equations Solve each equation. A) 6x + 3 - 8x = 13 6x + 3 - 8x = 13 Use the Commutative Property of Addition. Since 3 is added to -2x, subtract 3 from both sides to undo the addition. Combine like terms. 6x - 8x + 3 = 13 - 2x + 3 = 13 - 3 - 3 -2x = 10 Since x is multiplied by -2, divide both sides by -2 to undo the multiplication. -2x = 10 -2 -2 x = -5

11. CONFIDENTIAL 11 B) 9 = 6 - (x + 2) 9 = 6 - 2 - x 5 = -x Write subtraction as addition of the opposite. -5 = x Since 4 is added to -x, subtract 4 from both sides to undo the addition. 9 = 6 + (-1) (x + 2) 9 = 6 + (-1)(x) + (-1)(2) 9 = 6 - x - 2 9 = 4 - x 5 = -x -4 -4 Use the Commutative Property of Addition. Combine like terms. Since x is multiplied by -1, divide both sides by -1 to undo the multiplication. Simplify Distribute -1 on the right side.

12. CONFIDENTIAL 12 Now you try! 1) 2a + 3 - 8a = 8 2) -2 (3 - d) = 4 Solve each equation. Check your answer. 3) 4 (x - 2) + 2x = 40 3) x = 8 1)x = -5 6 2) d = 5

13. CONFIDENTIAL 13 Alex belongs to a music club. In this club, students can buy a student discount card for $19.95. This card allows them to buy CDs for $3.95 each. After one year, Alex has spent $63.40. Write and solve an equation to find how many CDs Alex bought during the year. Let c represent the number of CDs that Alex purchased. That means Alex has spent $3.95c. However, Alex must also add the amount spent on the card. total cost = cost of compact discs + cost of discount card i.e., 63.40 = 3.95c + 19.95 Make a Plan

14. CONFIDENTIAL 14 Solve 63.40 = 3.95c + 19.95 -19.95 43.45 = 3.95c 3.95 3.95 11 = c Alex bought 11 CDs during the year. The cost per CD is about $4, so if Alex bought 11 CDs, this amount is about 11(4) = $44. Add the cost of the discount card, which is about $20: 44 + 20 = 64. So the total cost was about $64, which is close to the amount given in the problem, $63.40. Check

15. CONFIDENTIAL 15 Sara paid $15.95 to become a member at a gym. She then paid a monthly membership fee. Her total cost for 12 months was $735.95. How much was the monthly fee? Now you try! $ 60

16. CONFIDENTIAL 16 Solving Equations to Find an Indicated Value If 3a + 12 = 30, find the value of a + 4. Since 12 is added to 3a, subtract 12 from both sides to undo the addition. Step1: Find the value of a. 3a + 12 = 30 -12 -12 3a = 18 Since a is multiplied by 3, divide both sides 3 to undo the multiplication. 3a = 18 3 Step2: Find the value of a + 4. a + 4 6 + 4 10 To find the value of a + 4, substitute 6 for a. Simplify.

17. CONFIDENTIAL 17 Now you try! 1) If 2x + 4 = -24, find the value of 3x. Solve each equation. Check your answer. 1) x = -42

18. CONFIDENTIAL 18 Assignments 1) 4a + 3 = 11 2) 8 = 3r - 1 3) 42 = -2d + 6 4) x + 0.3 = 3.3 Solve each equation. Check your answer: 1) a = 2 2) r = 3 3) d = -16 4) x = 3

19. CONFIDENTIAL 19 5) 15y + 31 = 61 6) 9 - c = -13 7) x + 4= 15 6 8) y + 1 = 5 3 4 12 9) If 3x - 13 = 8, find the value of x - 4. Solve each equation. Check your answer: 5) y = 2 6) c = 22 7) x = 66 8) y = 1 9) x - 4 = 5

20. CONFIDENTIAL 20 10) Paul bought a student discount card for the bus. The card cost $7 and allows him to buy daily bus passes for $1.50. After one month, Paul spent $29.50. How many daily bus passes did Paul buy? 10) 15

21. CONFIDENTIAL 21 Equations containing more than one operation can model real-world situations, such as the cost of a music club membership. Alex belongs to a music club. In this club, students can buy a student discount card for $19.95. This card allows them to buy CDs for $3.95 each. After one year, Alex has spent $63.40. To find the number of CDs c that Alex bought, you can solve an equation. $3.95c + $19.95 = $63.40 Cost of discount card Cost per CDTotal cost Next page Let’s review

22. CONFIDENTIAL 22 Notice that this equation contains multiplication and addition. Equations that contain more than one operation require more than one step to solve. Identify the operations in the equation and the order in which they are applied to the variable. Then use inverse operations and work backward to undo them one at a time. $3.95c + $19.95 = $63.40 Operations in the EquationTo Solve 1) First c is multiplied by 3.95. 2) Then 19.95 is added. 1) Subtract 19.95 from both sides of the equation. 2) Then divide both sides by 3.95. review

23. CONFIDENTIAL 23 Solving Two-Step Equations Solve 10 = 6 - 2x. Check your answer. 10 = 6 - 2x First x is multiplied by -2. Then 6 is added. - 6 -6 Work backward: Subtract 6 from both sides. 4 = -2x -2 -2 = x Since x is multiplied by -2, divide both sides by -2 to undo the multiplication. Check 10 = 6 - 2x 10 6 - 2 (-2) 6 - (-4) 10 review

24. CONFIDENTIAL 24 Solving Two-Step Equations That Contain Fractions Solve: q – 1 = 3 15 5 5 Method 1: Use fraction operations. q – 1 = 3 15 5 5 Since 1/5 is subtracted from q/5 15, add 1/5 to both sides to undo the subtraction. – 1 - 1 5 5 q = 4 15 5 15 × q = 15 × 4 = 60 15 5 5 Since q is divided by 15, multiply both sides by 15 to undo the division. Next page Simplify. q = 12 review

25. CONFIDENTIAL 25 Method 2: Multiply by the least common denominator (LCD) to clear the fractions. q – 1 = 3 15 5 5 Multiply both sides by 15, the LCD of the fractions. q – 1 = 3 15 5 5 15 × q – 1 = 3 15 5 5 15 × Distribute 15 on the left side. q - 3 = 9 + 3 + 3 q = 12 Simplify Since 3 is subtracted from q, add 3 to both sides to undo the subtraction. review

26. CONFIDENTIAL 26 Simplifying Before Solving Equations Solve each equation. A) 6x + 3 - 8x = 13 6x + 3 - 8x = 13 Use the Commutative Property of Addition. Since 3 is added to -2x, subtract 3 from both sides to undo the addition. Combine like terms. 6x - 8x + 3 = 13 - 2x + 3 = 13 - 3 - 3 -2x = 10 Since x is multiplied by -2, divide both sides by -2 to undo the multiplication. -2x = 10 -2 -2 x = -5 review

27. CONFIDENTIAL 27 B) 9 = 6 - (x + 2) 9 = 6 - 2 - x 5 = -x Write subtraction as addition of the opposite. -5 = x Since 4 is added to -x, subtract 4 from both sides to undo the addition. 9 = 6 + (-1) (x + 2) 9 = 6 + (-1)(x) + (-1)(2) 9 = 6 - x - 2 9 = 4 - x 5 = -x -4 -4 Use the Commutative Property of Addition. Combine like terms. Since x is multiplied by -1, divide both sides by -1 to undo the multiplication. Simplify Distribute -1 on the right side. review

28. CONFIDENTIAL 28 Alex belongs to a music club. In this club, students can buy a student discount card for $19.95. This card allows them to buy CDs for $3.95 each. After one year, Alex has spent $63.40. Write and solve an equation to find how many CDs Alex bought during the year. Let c represent the number of CDs that Alex purchased. That means Alex has spent $3.95c. However, Alex must also add the amount spent on the card. total cost = cost of compact discs + cost of discount card i.e., 63.40 = 3.95c + 19.95 Make a Plan review

29. CONFIDENTIAL 29 Solve 63.40 = 3.95c + 19.95 -19.95 43.45 = 3.95c 3.95 3.95 11 = c Alex bought 11 CDs during the year. The cost per CD is about $4, so if Alex bought 11 CDs, this amount is about 11(4) = $44. Add the cost of the discount card, which is about $20: 44 + 20 = 64. So the total cost was about $64, which is close to the amount given in the problem, $63.40. Check review

30. CONFIDENTIAL 30 Solving Equations to Find an Indicated Value If 3a + 12 = 30, find the value of a + 4. Since 12 is added to 3a, subtract 12 from both sides to undo the addition. Step1: Find the value of a. 3a + 12 = 30 -12 -12 3a = 18 Since a is multiplied by 3, divide both sides 3 to undo the multiplication. 3a = 18 3 Step2: Find the value of a + 4. a + 4 6 + 4 10 To find the value of a + 4, substitute 6 for a. Simplify. review

31. CONFIDENTIAL 31 You did a great job today!