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# Solving Equations by Multiplying or Dividing

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1. Solving Equations byMultiplying or Dividing CONFIDENTIAL

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

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

5. m= 1.5 3 4.5 3 1.5 1.51.5 B) m= 1.5 3 (3) × m= (3) × 1.5 3 Since m is divided by 3, multiply both sides by 3 to undo the division. m = 4.5 Check To check your solution, substitute 1.5 for m in the original equation. CONFIDENTIAL

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

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

8. 13 = -2w 13 -2(6.5) 1313 B) 13 = -2w 13 = -2w Since w is multiplied by -2, divide both sides by -2 to undo the multiplication. 13 = -2w -2 -2 -6.5 = w Check To check your solution, substitute 8 for x in the original equation. CONFIDENTIAL

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

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

12. B) 5= 4y 2 3 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 CONFIDENTIAL

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

14. (3)d = (3) × 10 3 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. Solution: Distance divided by 3 equals height in thousands of feet. d= h 3 Write an equation to represent the relationship. d= 10 3 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. CONFIDENTIAL

15. Properties of Equality 4 4 c c CONFIDENTIAL

16. Now you try! 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. 10x = 80 x = 10 milligrams CONFIDENTIAL

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

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

20. Let’s review 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. 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. CONFIDENTIAL

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

22. m= 1.5 3 4.5 3 1.5 1.51.5 review B) m= 1.5 3 (3) × m= (3) × 1.5 3 Since m is divided by 3, multiply both sides by 3 to undo the division. m = 4.5 Check To check your solution, substitute 1.5 for m in the original equation. CONFIDENTIAL

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

24. 13 = -2w 13 -2(6.5) 1313 review B) 13 = -2w 13 = -2w Since w is multiplied by -2, divide both sides by -2 to undo the multiplication. 13 = -2w -2 -2 -6.5 = w Check To check your solution, substitute 8 for x in the original equation. CONFIDENTIAL

25. review 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. CONFIDENTIAL

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

27. review B) 5= 4y 2 3 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 CONFIDENTIAL

28. (3)d = (3) × 10 3 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. Solution: Distance divided by 3 equals height in thousands of feet. d= h 3 Write an equation to represent the relationship. d= 10 3 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. CONFIDENTIAL

29. review Properties of Equality 4 4 c c CONFIDENTIAL

30. You did a great job today! CONFIDENTIAL