1. CONFIDENTIAL 1 Solving Equations by Multiplying or Dividing

2. CONFIDENTIAL 2 Warm Up 1) -5.2 + a = -8 Simplify: 2) 90 = -22 + a 3) -12 + f = 3 4) -1200 + j = 345 1) a = -2.8 2) a = 112 3) f = -9 4) j = 1545

3. CONFIDENTIAL 3 Introduction Solving an equation that contains multiplication or division is similar to solving an equation that contains addition or subtraction. Use inverse operations to undo the operations on the variable. Inverse Operations OperationInverse Operation MultiplicationDivision SubtractionAddition An equation is like a balanced scale. To keep the balance, whatever you do on one side of the equation, you must also do on the other side.

4. CONFIDENTIAL 4 Solve each equation. Check your answer. Solving Equations by Using Multiplication -4 = k -5 A) (-5) (-4) = (-5) ( k) -5 Since k is divided by -5, multiply both sides by -5 to undo the division. 20 = k Check -4 = k -5 -4 20 -5 To check your solution, substitute 20 for k in the original equation.

5. CONFIDENTIAL 5 m = 1.5 3 B) (3) × m = (3) × 1.5 3 m = 4.5 Check m = 1.5 3 1.5 4.5 3 Since m is divided by 3, multiply both sides by 3 to undo the division. To check your solution, substitute 1.5 for m in the original equation.

6. CONFIDENTIAL 6 Solve each equation. Check your answer. Now you try! 1) p = 10 5 2) -13 = y 3 3) c = 7 8 1) p = 50 2) y = -39 3) c = 56

7. CONFIDENTIAL 7 Solving Equations by Using Division Solve each equation. Check your answers. A) 7x = 56 7x = 56 7 7 Since x is multiplied by 7, divide both sides by 7 to undo the multiplication. Check 7x = 56 56 7(8)7(8) x = 8 To check your solution, substitute 8 for x in the original equation.

8. CONFIDENTIAL 8 B) 13 = -2w 13 = -2w -2 Since w is multiplied by -2, divide both sides by -2 to undo the multiplication. Check 13 = -2w -2(6.5) 13 -6.5 = w To check your solution, substitute 8 for x in the original equation.

9. CONFIDENTIAL 9 Now you try! 1) 16 = 4c 2) 0.5y = -10 Solve each equation. Check your answer. 3) 15k = 75 1) c = 4 2) y = -20 3) k = 5

10. CONFIDENTIAL 10 Remember that dividing is the same as multiplying by the reciprocal. When solving equations, you will sometimes find it easier to multiply by a reciprocal instead of dividing. This is often true when an equation contains fractions.

11. CONFIDENTIAL 11 Solving Equations That Contain Fractions Solve each equation. A) 5v = 35 9 The reciprocal of 5/9 is 9/5. Since v is multiplied by 5/9, multiply both sides by 9/5. 9 × 5v = 9 ×35 5 9 5 v = 63

12. CONFIDENTIAL 12 B) 5 = 4y 2 3 The reciprocal of 4/3 is 3/4. Since v is multiplied by 4/3, multiply both sides by 3/4. 3 × 5 = 3 ×4y 4 2 4 3 15 = y 8 5 = 4y 2 3

13. CONFIDENTIAL 13 Now you try! 1) -1 = b 4 5 2) 4j = 2 6 3 Solve each equation. Check your answer. 3) w = 102 6 1) b = -1.25 2) j = 1 3) w = 612

14. CONFIDENTIAL 14 The distance in miles from the airport that a plane should begin descending, divided by 3, equals the plane’s height above the ground in thousands of feet. If a plane is 10,000 feet above the ground, write and solve an equation to find the distance at which the pilot should begin descending. Distance divided by 3 equals height in thousands of feet. Solution: d = h 3 Write an equation to represent the relationship. Substitute 10 for h. Since d is divided by 3, multiply both sides by 3 to undo the division. d = 30 The pilot should begin descending 30 miles from the airport. d = 10 3 (3)d = (3) × 10 3

15. CONFIDENTIAL 15 Properties of Equality WORDSNUMBERSALGEBRA Multiplication Property of Equality You can multiply both sides of an equation by the same number, and the statement will still be true. 6 = 6 6 (3) = 6 (3) 18 = 18 a = b ac = bc Division Property of Equality You can divide both sides of an equation by the same nonzero number, and the statement will still be true. 8 = 8 2 = 2 a = b (c ≠ 0) a = b 44 cc

16. CONFIDENTIAL 16 An orange contains about 80 milligrams of vitamin C, which is 10 times as much as an apple contains. Write and solve an equation to find the amount of vitamin C in an apple. Now you try! 10x = 80 x = 10 milligrams

17. CONFIDENTIAL 17 Assignments 1) K = 8 4 2) z = -9 3 3) -2 = w -7 4) 6 = t -5 Solve each equation. Check your answer: 1) k = 32 2) z = -27 3) w = 14 4) t = - 30

18. CONFIDENTIAL 18 5) g = 10 1.9 6) 2.4 = b 5 7) 4x = 28 8) -64 = 8c 9) -9 j = -45 Factor the trinomial. Check your answer. 5) g = 19 6) b = 12 7) x = 7 8) c = - 8 9) j = 5

19. CONFIDENTIAL 19 10) The Baseball Birthday Batter Package at a minor league ballpark costs $192. The package includes tickets, drinks, and cake for a group of 16 children. Write and solve an equation to find the cost per child. 10) 16 c =$192 c = $12

20. CONFIDENTIAL 20 Solving an equation that contains multiplication or division is similar to solving an equation that contains addition or subtraction. Use inverse operations to undo the operations on the variable. Inverse Operations OperationInverse Operation MultiplicationDivision SubtractionAddition An equation is like a balanced scale. To keep the balance, whatever you do on one side of the equation, you must also do on the other side. Let’s review

21. CONFIDENTIAL 21 Solve each equation. Check your answer. Solving Equations by Using Multiplication -4 = k -5 A) (-5) (-4) = (-5) ( k) -5 Since k is divided by -5, multiply both sides by -5 to undo the division. 20 = k Check -4 = k -5 -4 20 -5 To check your solution, substitute 20 for k in the original equation. review

22. CONFIDENTIAL 22 m = 1.5 3 B) (3) × m = (3) × 1.5 3 m = 4.5 Check m = 1.5 3 1.5 4.5 3 Since m is divided by 3, multiply both sides by 3 to undo the division. To check your solution, substitute 1.5 for m in the original equation. review

23. CONFIDENTIAL 23 Solving Equations by Using Division Solve each equation. Check your answers. A) 7x = 56 7x = 56 7 7 Since x is multiplied by 7, divide both sides by 7 to undo the multiplication. Check 7x = 56 56 7(8)7(8) x = 8 To check your solution, substitute 8 for x in the original equation. review

24. CONFIDENTIAL 24 B) 13 = -2w 13 = -2w -2 Since w is multiplied by -2, divide both sides by -2 to undo the multiplication. Check 13 = -2w -2(6.5) 13 -6.5 = w To check your solution, substitute 8 for x in the original equation. review

25. CONFIDENTIAL 25 Remember that dividing is the same as multiplying by the reciprocal. When solving equations, you will sometimes find it easier to multiply by a reciprocal instead of dividing. This is often true when an equation contains fractions. review

26. CONFIDENTIAL 26 Solving Equations That Contain Fractions Solve each equation. A) 5v = 35 9 The reciprocal of 5/9 is 9/5. Since v is multiplied by 5/9, multiply both sides by 9/5. 9 × 5v = 9 ×35 5 9 5 v = 63 review

27. CONFIDENTIAL 27 B) 5 = 4y 2 3 The reciprocal of 4/3 is 3/4. Since v is multiplied by 4/3, multiply both sides by 3/4. 3 × 5 = 3 ×4y 4 2 4 3 15 = y 8 5 = 4y 2 3 review

28. CONFIDENTIAL 28 The distance in miles from the airport that a plane should begin descending, divided by 3, equals the plane’s height above the ground in thousands of feet. If a plane is 10,000 feet above the ground, write and solve an equation to find the distance at which the pilot should begin descending. Distance divided by 3 equals height in thousands of feet. Solution: d = h 3 Write an equation to represent the relationship. Substitute 10 for h. Since d is divided by 3, multiply both sides by 3 to undo the division. d = 30 The pilot should begin descending 30 miles from the airport. d = 10 3 (3)d = (3) × 10 3

29. CONFIDENTIAL 29 Properties of Equality WORDSNUMBERSALGEBRA Multiplication Property of Equality You can multiply both sides of an equation by the same number, and the statement will still be true. 6 = 6 6 (3) = 6 (3) 18 = 18 a = b ac = bc Division Property of Equality You can divide both sides of an equation by the same nonzero number, and the statement will still be true. 8 = 8 2 = 2 a = b (c ≠ 0) a = b 44 cc review

30. CONFIDENTIAL 30 You did a great job today!